Conical Refraction

Chapter XIII of Life of Sir William Rowan Hamilton (Vol. 1, Hodges Figgis, Dublin, 1882).

By Robert Perceval Graves.

In one of Hamilton's letters to Lord Adare he speaks of having copied out his Third Supplement to his Theory of Systems of Rays as many as ten times, in the endeavour to perfect it; but this letter does not record that, while thus giving final shape to his work, he had arrived by means of his general method at an optical result of a most remarkable nature. This, however, was the fact. He had made the theoretical discovery of Conical Refraction. And when he presented the concluding part of his Third Supplement to the Royal Irish Academy on the 22nd of October, 1832, it contained a statement of the discovery, which he then orally announced. Of the position in optical science of this discovery, the unscientific reader will gather a correct notion from the following passage which I reproduce from the memoir published in the Dublin University Magazine of January, 1842.

`The law of the reflexion of light at ordinary mirrors appears to have been known to Euclid; that of ordinary refraction at a surface of water, glass, or other uncrystallized medium, was discovered at a much later age by Snellius; Huyghens discovered, and Malus confirmed, the law of extraordinary refraction produced by uniaxal crystals, such as Iceland spar; and finally the law of the extraordinary double refraction at the faces of biaxal crystals, such as topaz or arragonite, was found in our own time by Fresnel. But even in these cases of extraordinary or crystalline refraction, no more than two refracted rays had ever been observed or even suspected to exist, if we except a theory of Cauchy, that there might possibly be a third ray, though probably imperceptible to our senses. Professor Hamilton, however, in investigating by his general method the consequences of the law of Fresnel, was led to conclude that there ought to be in certain cases, which he assigned, not merely two, nor three, nor any finite number, but an infinite number, or a cone of refracted rays within a biaxal crystal, corresponding to and resulting from a single incident ray; and that in certain other cases, a single ray within such a crystal should give rise to an infinite number of emergent rays, arranged in a certain other cone. He was led, therefore, to anticipate from theory two new laws of light, to which he gave the names of Internal and External Conical Refraction.'

So sure was Hamilton's grasp of his mathematical results, and of the necessary correspondence with them of physical phenomena (the truth of the undulatory theory being supposed), that on the day succeeding the above-mentioned meeting of the Royal Irish Academy, he requested his friend Mr. Lloyd, afterwards Provost of Trinity College, Dublin, and then Professor of Natural Philosophy, to institute experiments for the purpose of verifying his theoretical anticipations. The task was promptly undertaken, and besides the letters between Hamilton and Lloyd which record its progress, others from Hamilton to Airy and Herschel, with their replies, are in existence, which are of great interest. Being full of mathematical formulae, they are more suited for a collection of the scientific correspondence of the subject of this memoir, which I hope may some day see the light, than for the present work. Here it must suffice to give an outline of their contents, indicating the history of the discovery and its verification, and one or two letters of general statement.

The earliest letter of the series which remains is Hamilton's reply to Lloyd's inquiry respecting the angle of the cone, for arragonite, in the case of external conical refraction. It commences thus:-

`November 3, 1832, Saturday morning. - Mrs. Hemans and some of the young Graveses came here yesterday evening, just as I had finished my calculation respecting the arragonite, and I had only time to write as answer, ``3°.'' [1] I showed the cabalistic note to Mrs. Hemans, and she admitted that we professors had attained the perfection of letter-writing.'

He then enters upon a consideration of some of Lloyd's observations in comparison with his own results of theoretical calculation, and prepares Lloyd for finding that the cone would not be exactly circular. On the following day he suggests to Lloyd an easy experimental verification by means of a slit in a card.

On the 25th of October, Hamilton had written to Airy, offering to propose him as an Honorary member of the Royal Irish Academy, and stated in general terms that he had arrived at some new results from Fresnel's theory. On the 4th November Airy replies -

`I am much obliged by your note of October 25. I should highly value the honour of being a member of your Academy, and I should esteem it much more because it originated with you.... I shall be glad in time to hear of the new results of Fresnel's theory which you allude to.'

On the 6th November, Lloyd reports some unsuccessful experiments, and his intention to try another way of his own devising, and also that suggested by Hamilton, but concludes by saying, `I almost despair of doing anything with so thin a plate [of arragonite].'

On the 10th of November Hamilton writes thus to Lloyd:-

`Just after the evening when I gave to the Royal Irish Academy an account of my last optical results, I wrote to Professor Airy, and among other things I mentioned that I had arrived at a new consequence from Fresnel's theory, without stating what that consequence was. I now enclose a letter received from him yesterday, in which he expresses a wish to be informed of it; and if you should, as you seemed to think likely, be prevented by want of apparatus or of leisure from making soon any decisive experiment on the point, I believe it will be well to mention the theoretical result to Airy.'

To this letter Professor Lloyd sent the following reply. It is impossible not to be struck by the pure unselfish zeal for science which it displays.

`I fear it would be wholly impossible to obtain experimentally any decisive result connected with your theoretical conclusion, without better means than I have at present at my disposal. The angle of divergence produced by diffraction in the minutest apertures, when they are so close as they must be in my specimen, is far greater than the angle we seek. The specimens I showed you the other day are fine, but I find they belong to a form of crystallization which the mineralogists term macled, that is, in fact, they are composed of several distinct crystals crossing each other. They would be therefore wholly unfit for the purpose. I am quite sure your conclusion can be readily tested by anyone having access to fair specimens; but as that is not the case here, you had better refer the matter to Airy, or some one else, as soon as possible.''

But happily the honour of bringing these experiments to a successful termination was not to pass from Professor Lloyd. Within a few days he had procured a better specimen of the required crystal, and he has the pleasure of thus writing to Hamilton:

`Trinity College, December 14. - Dear Hamilton, I write this line to say that I have found the cone. At least I have almost no doubt on the subject; but must still verify it by different methods of observation.

`I have no time to say more at present than that I observed it in a fine specimen of arragonite which I received from Dollond in London since I saw you last.'

On the 18th of December Hamilton communicated this verification of his theoretical anticipation to both Airy and Herschel. I give a transcript of his letter to the last; it is an interesting though not a full statement of the discovery and the verification.

From W. R. Hamilton to Sir J. F. W. Herschel.

`Dublin Observatory, December 18, 1832.

`You are aware that the fundamental principle of my optical methods does not essentially require the adoption of either of the two great theories of light in preference to the other. However I naturally feel an interest in applying my general methods to Fresnel's theory of biaxal crystals; and when in October I was finishing my Third Supplement for the Royal Irish Academy, I deduced, from such application, some results respecting the focal lengths and aberrations of lenses formed of such crystals. In the course of these calculations I was led to transform in various ways Fresnel's law of velocity, or, in other words, to study his curved wave: and I found, what he seems to have not suspected, that the wave has 1st, four cusps (at the ends of the optic axes) at each of which the tangent planes are (not, as he thought, two, but) infinite in number; and 2nd, four circles of plane contact, along each of which the wave is touched, in the whole extent of the circle, by a plane (parallel to one of the circular sections of the surface of elasticity); somewhat as a plum can be laid down on a table so as to touch and rest on the table in a whole circle of contact, and has, in the interior of the circular space, a sort of conical cusp. Hence I was led to expect that under certain circumstances, easily deduced and assigned by me from these geometrical properties, a single incident and unpolarized ray would undergo not double but conical refraction. I announced this expectation to the Royal Irish Academy at their monthly meeting in October, when I was giving an account of the results of my Third Supplement; and I applied to Professor Lloyd, son of our Provost here, to submit the matter to experiment. For some time he could do nothing decisive, not having any biaxal crystal of sufficient size and purity; but having lately obtained from Dollond a fine piece of arragonite, and having treated it according to my theoretical indications, he has perceived a curious and beautiful set of new phenomena, which, so far as they have yet been examined, appear to agree with the theory, and at any rate are worthy of study. I thought this intelligence would interest you, and I am,' &c.

On the same day Lloyd writes to Hamilton as follows:-

`Tuesday, 3 o'clock; College. - I am happy to tell you that since I saw you this morning I succeeded in projecting the cone on a screen of roughened glass, and observing a section of it so large as two inches in diameter; you will easily conceive that the phenomenon is most striking. The appearance is exactly the same as that we saw when looking through the aperture. Its deviation from an exact circle, however, is of course more distinctly seen. I traced the boundary of this section on the screen, and then measured the distance as accurately as I could. Three such measurements gave me for the angle of the cone 6° 24´, 6° 22´, 5° 56´, which you see are tolerably near. The mean (6° 14´) corresponds pretty well with the measurements of the extreme circle, taken yesterday. The difference between it and the theoretical result is probably the effect of diffraction, and I must now try and correct for this perturbation. This mode of exhibiting the phenomenon is decisive as well as beautiful, and I am sure you will be glad to see it when you next come in to town.'

On the 23rd of December, Professor Airy writes:-

`I have duly received your letter concerning double refraction, and that informing me of my election as Honorary Member of the Royal Irish Academy (of which I had not received an official notice). I beg you to say to the authorities of that Body that I am very much gratified with the honour which they have done me, and I hope it may prove the cause of greater personal acquaintance with many of its members than I at present possess.

`I am very much interested with your discovery of the circular contact of the tangent plane with Fresnel's double wave surface. I was well aware (a long time ago) that the point of the surfaces, which in the principal section is the intersection of the circle and the ellipse, is in the surfaces the meeting of two dimples (external and internal), and that these dimples near their point of meeting become ultimately two opposite cones; the outer one diverging in a sort of trumpet mouth. But I had no idea that the mouth of the trumpet could be touched by one plane. Now as to the consequences of this I am extremely puzzled.... Arragonite is a bad substance, I should imagine; I should think topaz likely to make a wider cone; [2] perhaps your formulae will show you at once. Let me beg you to communicate as soon as possible (if Professor Lloyd does not object) the phenomena which he has observed. I have to thank him for a copy of his excellent optical treatise.'

I regret that I have not been able to find the letter from Professor Lloyd and its enclosure (presumably a note from Mr. Mac Cullagh, F.T.C.D.), to which the following important letter is a reply.

From W. R. Hamilton to Professor Lloyd.

`Observatory, January 1, 1833.

`I have just received your letter and the enclosed note and write in some haste. Mr. Mac Cullagh's last conclusion that the conical refraction at emergence required the internal ray along the optic axis to be unpolarized, or to be formed by the superposition of rays polarized in infinitely various planes, is exactly the same with the conclusion which I had formed in October, and I distinctly remember mentioning it to you in our interview on the 23rd of that month; and it was for that reason I wished to have the luminous point in contact with the crystal. But I have not yet tried to determine the exact law connecting the internal plane of polarization of an internal polarized ray with the position of the corresponding emergent ray of the cone, though the determination will not be difficult, and the result probably very nearly the same as in that other connected question [3] which we talked of the other day, and which we had both resolved by different methods. What has hindered me from setting about this little problem has been my being much engaged and interested in Cauchy's theory of light. [4] As to the finite magnitude of the emergent cone, for a single internal common ray, I certainly expect a finite magnitude, that is, a finite angle (though the cone of rays is not of revolution nor even of the second degree), but not a conical shell of finite thickness, such as one may consider as approximately resulting from a finite but thin internal cylinder of rays. As this last is nearly the case of the experiment, there must no doubt be a thickness in the cone of the order of the aperture besides the angular divergence; and this may, as you say, account for part of what you observed, but I scarcely think it will account for the whole. It is much for theory to have predicted the facts of conical refraction, but I suspect that the exact laws of it depend on things as yet unknown. You see my pleasure at perceiving so great a confirmation of theory does not make me sanguine enough to believe as yet the coincidence absolute and rigorous. As to rays inclined a little to the optic axis all round, it was in fact from considering them and passing to the limit that I first deduced my expectation of conical refraction. When you are drawing up your Paper I shall be glad if your plan leads you (when you are speaking of my having requested you to try experiments) to mention distinctly the following facts, which constitute all my merit, such as it is, on the subject.

  1. `I announced to you on the 23rd of October last, having on the preceding evening announced to a general meeting of the Royal Irish Academy, that I had discovered two new geometrical properties of Fresnel's wave; one property being the existence of four conoidal cusps at the intersections of circle and ellipse in the plane of the greatest and least axes; and the other property being the existence of four finite circles of plane contact, each of the four planes of these circles being parallel to one of the two circular sections of the surface of elasticity.
  2. `I announced to you on the same day, and had done so to the Academy on the evening before, my expectation of a new kind of refraction, namely conical refraction, which ought to happen in two distinct cases; one at emergence, when a single ray of light from a point within a biaxal crystal proceeded along an optic axis (from centre to cusp of Fresnel's wave) and then emerged; the other at entering when a single ray of common light from a point without falls on a biaxal crystal and enters so that the plane wave within, or the tangent plane to the curved wave within, is parallel to either of the two circular sections of the surface of elasticity.
  3. `I requested you to try experiments to confirm or refute the theoretical expectations which I had deduced from Fresnel's principles.

`You intended I know to mention the third, but you might not have thought of distinctly putting the two others on record, which yet may save some controversy with others hereafter. I expect on Thursday evening to leave the neighbourhood of Dublin for a few days, but to return early next week.'

That Hamilton was ready to make known the work done by Mac Cullagh in the same field with himself is proved by the following passage in a letter of his to Professor Airy, written, as a shorthand draft of it shows, a few days later, viz., January 4, 1833.

`I hear from Lloyd that Mac Cullagh (another of our young Fellows, a Paper by whom I once showed you) had deduced the same results by his geometrical methods, having however previously heard of my theory of conical refraction.' [5]

This letter to Airy communicated at length the results of Professor Lloyd with respect to external conical refraction, together with some views of Hamilton's own as to the `vibrations', `interference,' and `polarization,' involved in the experiment.

In Professor Airy's answer, after referring to polarization, he expresses strongly his conviction that if the phenomenon of external conical refraction be true in fact, it has no connexion with the theory of Hamilton. He then ably sketches what he considers possible results, but shows that he has misconceived Hamilton's statement.

To this letter Hamilton sent a reply on the 21st of January, and, not hearing in return from Airy, another on the 1st of February; in these letters he modestly, and it may be in accordance with the fact, supposes that some ambiguity in his own expressions may have caused his correspondent's failure correctly to appreciate the results arrived at by himself and Professor Lloyd, [6] and after re-stating and explaining them he quietly adds, `I believe that if you consider the thing you will come to the same conclusion with me.' On the very day on which he had despatched the second of these letters Hamilton received from Professor Airy a letter dated January 28, which handsomely acknowledged that he had been convinced by Hamilton's explanation; the following are its terms:-

From Professor Airy to W. R. Hamilton.

`Allow me to thank you for your last note, which is all comprehensible and all true; and if I had not been very dull, I might perhaps have guessed at some of it before. You had not mentioned to me anything about the cusp-ray, and therefore there were parts of the previous letter which were altogether mysterious to me, and were likely to remain so, except I could divine or you explain.'

It will be seen [7] that, not long after, Professor Airy followed up this private amende by a public testimony, still fuller, though couched in fewer terms, to the character of Hamilton's discovery as a scientific feat.

The following letter of this date to Herschel is so clear a statement of almost everything connected with this discovery that I feel I ought not to suppress it, though aware that its production involves some repetition.

From W. R. Hamilton to Sir John Herschel.

`Observatory, January 29, 1833.

`My dear Sir,

`Professor Lloyd read to the Royal Irish Academy, last night, a paper ``On the Phenomena presented by Light in its Passage along the Axes of Biaxal Crystals,'' in which he gave an account of some recent additional experiments, confirming my theoretical conclusions respecting Conical Refraction. Those conclusions were chiefly the following:- 1. A single plane wave within a biaxal crystal, parallel to a circular section of the surface of elasticity, corresponds in general to an infinite number of internal ray-directions; in such a manner that a single incident ray in air will give an internal cone of rays (of the 2nd degree), and will emerge (from a plane face) as an external cylinder of rays, if the external incident wave have that direction which corresponds to the foregoing internal wave. In this kind of internal conical refraction one refracted ray of the cone is determined by the ordinary law of the sines, using the mean index 1/b; and the greatest angular deviation in the cone, from this ray, is in the plane of the optic axes, and is

= \tan^{-1} \frac{\sqrt{a^2 - b^2} \sqrt{b^2 - c^2}}{b^2} = 1^\circ 55'
for ray E in arragonite, if we use Rudberg's elements. Professor LLoyd has lately observed an emergent cylinder corresponding to this theory, from his measures upon which the angle of the cone appeared to be 1° 52´. He used a fine piece of arragonite, procured from Dollond, thickness = 0.49 inch; the incident ray was of solar light, and it passed through two small holes, the first in a screen at some distance from the crystal, the second in a thin metallic plate, adjoining the first surface of the crystal; the emergent cylinder of rays was received on silver paper, and produced on the paper a small white annulus of which the size was the same at different distances of the paper from the arragonite. The emergent light was polarized according to a law which agrees with Fresnel's principles. Great care was necessary in the adjustment of the holes; when the adjustment was slightly disturbed, two opposite quadrants of the circle appeared more faint than the two others, and the two pairs were of complementary colours.

2. `I conclude also, from Fresnel's principles, that a single interior cusp-ray (often called an optic axis, but not normal to a circular section of the surface of elasticity, and on the contrary normal to a circular section of Fresnel's ellipsoid - one of those two rays of which each has but a single value for the velocity of light along it -) ought, on emerging into air, to undergo, not bifurcation as Fresnel thought, but (external) conical refraction. If the internal incidence be perpendicular, the equation in rectangular co-ordinates of the emergent cone may be put under the form

\frac{x^2 + y^2}{x\sqrt{x^2 + y^2 + z^2}}
   = \frac{\sqrt{a^2 - b^2} \sqrt{b^2 - c^2}}{abc} = \sin 2^\circ 57'
for ray E, with Rudberg's elements for arragonite; this cone therefore is of the 4th degree (whereas the internal cone was of the 2nd), but it does not differ much from a circular cone. In Professor Lloyd's experiments the normal to the refracting face was Fresnel's axis a, bisecting the acute angle between the two cusp-rays, and the internal incidence was about 10°; which made the theoretical angle of the emergent cone somewhat more than 3° instead of 2° 57´. He has sent to the Annals of Philosophy [8] a sketch of his experimental results which appear to agree sufficiently with the theory, as to the position and magnitude and polarization of the emergent cone, in this external conical refraction. More lately he has taken new measures which appear to agree still better; and he has made those experimental verifications, which I have attempted in this letter to describe, of the other (the internal) kind of conical refraction. The appearances in direct vision, or when the light is received on a screen, are interesting enough, and vary prettily with the shape and size of the aperture, in the phenomena of external conical refraction. Figures will be given in the fuller memoir in the Transactions of our Irish Academy.

`The experimental establishment of these new consequences from Fresnel's principles, must, I think, be considered as interesting. My Third Supplement, in which, besides endeavouring in other ways to perfect my optical methods, I treat of the connexion of my mathematical view with the undulatory theory of light, is in the press, but gets on very slowly. Whenever it is printed, which can scarcely be in less than two or three months, I shall present you with a copy. Meanwhile believe me,' &c.

In the February and March numbers of the London and Edinburgh Philosophical Magazine, pp. 112 and 207, were contained two Papers giving Professor Lloyd's earliest published account of his experiments, the first of them describing external, the second, internal conical refraction. They prove that Hamilton was fortunate in his coadjutor. The conduct of the experiments called for much ingenuity in devising physical arrangements and the utmost nicety of observation; and these Papers furnish full evidence of the exercise of both by Professor Lloyd. They show also that he was more than a mere verifier; he took note of a phenomenon that had not been predicted, and ascertained the law to which it conformed. When investigating the case of external conical refraction, he discovered, by observation with a tourmaline plate, that all the rays of the cone were polarized in different planes, and detected the remarkable law that `the angle between the planes of polarization of any two rays of the cone is half the angle contained by the planes passing through the rays themselves and its axis': this law he also proved to be a necessary consequence of Fresnel's theory. Upon the phenomenon being communicated to Hamilton, he likewise, by means of his own methods, deduced the same law from the theory, and subsequently predicted the corresponding phenomenon in the case of internal conical refraction together with its analogous law. In this latter case, the prediction of the phenomenon and its law received its experimental verification at the hands of Professor Lloyd: in the former case, it has been seen, he had observed the unpredicted phenomenon, and had preceded Hamilton in deducing its law from theory. It has become necessary thus distinctly to put on record the amount of credit due to Professor Lloyd in this particular, because it has been overlooked by Professor Tait in the lucid account of the discovery which is contained in his article on Hamilton in the North British Review of September, 1866. The omission arose very naturally from the circumstance that these laws for polarization in both kinds of conical refraction are given in Hamilton's Third Supplement, which was communicated to the Academy previously to Lloyd's researches, but which remained unpublished (as appears from the Introduction) for many subsequent months. The correspondence in my hands proves that the part of the paper concerning polarization must have been inserted at a date subsequent to the 2nd January, 1833. But that Hamilton was willing to leave with Lloyd the credit of the priority which has been here assigned to him is proved by the fact, which I have received on the best authority, that he requested and obtained permission to circulate the private copies of his friend's paper (in which the above-mentioned facts are recorded) along with those of his own memoir.

In the XVIIth volume of the Transactions of the Royal Irish Academy, Part I., which was published in the summer of 1833, may be found both Hamilton's Third Supplement, containing his theoretical discovery of Conical Refraction, and Professor Lloyd's perfected account of the experimental manifestations of both kinds of it, accompanied by plates of diagrams representing the phenomena. To these Papers the scientific reader is referred for full information on the subject. They link the names of Hamilton and Lloyd in an enduring bond.

I may fitly conclude this statement by again borrowing a passage from the memoir of Hamilton, published in the Dublin University Magazine for January, 1842:-

`This result excited at the time a very considerable sensation among scientific men in England and on the Continent; it was thought a happy boldness to have thus seized and brought forth into view, by dint of reasoning, a new class of phenomena, to which nothing similar had been before observed, and which even seemed, in the words used by an eminent English philosopher, to be ``in the teeth of all analogy.'' At the Cambridge meeting of the British Association, in 1833, the attention of the mathematical and physical section was largely given to the subject, and Herschel, Airy and others, spoke warmly in praise of the discovery. In the introductory discourse with which the proceedings of that meeting were opened, Professor Whewell made it a topic, and expressed himself in the following words:- ``In the way of such prophecies, few things have been more remarkable than the prediction, that under particular circumstances a ray of light must be refracted into a conical pencil, deduced from the theory by Professor Hamilton, and afterwards verified experimentally by Professor Lloyd.'' [9] Previously, in the same year, Professor Airy had publicly recorded his impression upon the subject as follows:- ``Perhaps the most remarkable prediction that has ever been made is that lately made by Professor Hamilton.'' [10] More lately, Professor Plücker, of Bonn, in an article on the general form of luminous waves, published in the nineteenth volume of Crelle's Journal, has used these words:- ``Aucune experience de physique a fait tant d'impression sur mon esprit, que celle de la refraction conique. Un rayon de lumière unique entrant dans un crystal et en sortant sous l'aspect d'un cone [11] lumineux: c'etait une chose inouie et sans aucune analogie. Mr. Hamilton l'annonça, en partant de la forme de l'onde, qui avait été deduite par des longs calculs d'une theorie abstraite. J'avoue que j'aurois désesperé de voir confirmé par l'experience un résultat si extraordinaire, prédit par la seule théorie que las genie de Fresnel avait nouvellement créée. Mais Mr. Lloyd ayant démontré que les experiences etaient en parfaite concordance avec les predictions de Mr. Hamilton, tout préjugé contre une théorie si merveilleusement soutenue, a dû disparaitre.'' And it seems to be in part to this subject that reference is made in a passage of the article, attributed to Sir John Herschel, on the Inductive Sciences, in the number for last June [1841] (p. 233) of the Quarterly Review, where mention is made of ``a sound induction enabling us to predict, bearing not only stress, but torture: of theory actually remanding back experiment to read her lesson anew; informing her of facts so strange, as to appear to her impossible, and showing her all the singularities she would observe in critical cases she never dreamed of trying.'' '

In the Bridgewater Treatise of Mr. Babbage the author not only bears his testimony to the merits of Hamilton and Lloyd, but manifests his appreciation of the remarkable character of the discovery by weaving it as a typical example into the argument of his book. It has more recently been characterized as in its own sphere to be classed with that prediction of the existence of the planet Neptune which has immortalized the names of Adams and Le Verrier. Yet it will be seen by his letter to Coleridge of February 3, 1833, that Hamilton himself looked upon this and all similar predictions as `a subordinate and secondary result,' when compared with the object he had in view, - `to introduce harmony and unity into the contemplations and reasoning of optics, regarded as a branch of pure science.'

Note on Conical Refraction: Hamilton and Mac Cullagh

As I find that the relative positions of Hamilton and Professor Mac Cullagh in regard to the discovery of Conical Refraction are still, from time to time, matter of discussion, I feel it necessary to add as a note the following statement:-

To the August number of the Philosophical Magazine for 1833 (p. 114) was communicated a Paper by Mr. Mac Cullagh entitled, `Note on the subject of Conical Refraction,' which commences with the following paragraphs:-

`When Professor Hamilton announced his discovery of Conical Refraction, he does not seem to have been aware that it is an obvious and immediate consequence of the theorems published by me, three years ago, in the Transactions of the Royal Irish Academy, vol. xvi., part ii., p. 65, &c. The indeterminate cases of my own theorems which, optically interpreted, mean conical refraction, of course occurred to me at the time, but they had nothing to do with the subject of that Paper; and the full examination of them, along with the experiments they might suggest, was reserved for a subsequent essay, which I expressed my intention of writing. Business of a different nature, however, prevented me from following up the inquiry.

`I shall suppose the reader to have studied the passage in pp. 75, 76, of the volume referred to. he will see that when the section of either of the two ellipsoids employed there is a circle, the semiaxes - answering to OR, Or and to OQ, Oq, in the general statement - are infinite in number, giving of course an infinite number of corresponding rays. And this is conical refraction.'

The note then gives geometrical deductions from his previously published geometrical theorems which correspond with the two cases of conical refraction.

Hamilton was hurt by the terms in which the first of these paragraphs was couched; he meditated a reply to it, and informed Professor Lloyd of his intention. From the latter, early in the month, he received the following reply:-

From Professor Lloyd to W. R. H.

`Killiney, August 9, 1833.

`Shortly after I left you on Thursday last I met Mac Cullagh, and thought it better to avail myself of the liberty you allowed me, and mentioned that you were about to answer his note. I did not enter further into the subject, but in the few words which followed he mentioned that he had explicitly stated to you, at the time of his first publication, his intention of writing a supplemental essay on Fresnel's Theory, and that he had made a similar communication to my father.

`I took no further notice of this at the time, but on my return to the country I though it would save much embarrassment and recrimination to make you aware of this fact, which probably has escaped your recollection. I therefore wrote a short note to Mac Cullagh, yesterday morning, to inquire whether it was to this he referred in the passage in his last not, on which you have dwelt so much in your reply, and to ask permission, if it were so, to state the fact to you. I received last night his distinct affirmation to both these points, and along with it some further details which lead me to hope that the matter may be adjusted in a less hostile manner. In this hope I now write to urge you to take no further step in this matter until I see you. I shall be in town on Monday morning, when you will probably come in to attend the Academy, if not for this business, which I cannot but regard as of much importance both to you and Mac Cullagh. I trust I shall then be able to adjust the matter to the satisfaction of both parties; but if not, it will not be too late for you to persevere in your present intention of a reply.'

To this note Hamilton briefly replies on the same day:-

From W. R. Hamilton to Professor Lloyd.

`Observatory, August 10, 1833.

`It is very friendly in you to take so much trouble about the matter, and what you state in your last note is very important. It has quite escaped my recollection that Mac Cullagh mentioned to me any intention of writing a supplemental essay on Fresnel; but of course I do not doubt his word. I still think I ought to state distinctly that I was (until very lately) under the impression that he had not in any degree anticipated me, and that he lately mentioned to me that he had suppressed his own expectations. But certainly I am anxious not to appear nor to be hostile to him; and I fully intend to be at the Academy on Monday next, in the hope of meeting you and him, if you think it well to do so.'

Later in the month, on the 22nd, Lord Adare writes to Hamilton as follows:-

`Dear Professor, I hear Mac Cullagh has published in the Phil. Mag. a Paper in which he says he had arrived at Conical Refraction some time ago. Of course this will not pass without some remarks from you.'

In answer, Hamilton gives his friend the following interesting account of what had been passing:-

From W. R. H. to Viscount Adare.

`Observatory, August 29, 1833.

`When I saw Mac Cullagh's remarks in the Phil. Mag. for this month, I was certainly a little offended, for they seemed to insinuate that I might have got the hint from his Memoir; and I amused myself writing an answer in a somewhat satirical vein. But I took the precaution of showing it to Professor Lloyd, who, on receiving it, immediately came here in great alarm lest Mac Cullagh and I should get into an unpleasant controversy. I asked Lloyd, but this of course is entre nous, whether he really thought from his long acquaintance that Mac Cullagh was an honest man; and he assured me that he had the highest opinion of his honour. He said, too, that Mac Cullagh had lately brought some things to his recollection which agreed with Mac Cullagh's recent statements of his having thought something odd would arise in connexion with the circular sections of the two ellipsoids in the theory, though he did not communicate his thoughts to others, nor develop them himself. In particular, Lloyd remembers that Mac Cullagh complained to him some years ago, that on his asking a Dublin optician for crystals, he was shown the crystal of a watch. But Mac Cullagh did not then tell Lloyd what he wanted the crystals for, nor (so far as I can learn) had he any distinct expectation himself. However, Lloyd's assurances of his confidence in Mac Cullagh's honour changed a good deal my state of feeling; though I still thought of writing to the Magazine, and indeed Lloyd himself said that some unguarded expressions in Mac Cullagh's remarks required some notice to be taken of them. But before the time expired within which I should have written, if at all, Lloyd brought me a message from Mac Cullagh that he was very sorry for having unintentionally offended me; that the obnoxious sentences were written in great haste, to save the post and the month, and were sent later than the body of his little Paper (though they are printed at the beginning), under the influence of a friend who urged him to make some claim, which he had not at first intended to do, but merely to deduce geometrically the two cases of conical refraction from his own theorems and methods; and finally that he was willing to publish in the next number of the Magazine an explanation, a copy of which was shown me, containing a statement that he had not only not communicated his thoughts to others, but had not perfectly developed them himself; until by hearing of my results he was led to resume the inquiry, and to deduce the demonstrations which he gave in the last number. You will easily suppose that I was quite pacified by this, and thought it needless to indulge the world with the spectacle of a battle between us, which would no doubt have furnished rare entertainment.

`When all was over, I thanked Lloyd for the trouble he had taken, and hinted that having reconciled us it would be well not to mention to Mac Cullagh the doubts which I felt for a while with respect to his truth and honour. He laughed at this, and said, that would indeed be drawing the line upon the crystal, in allusion to one of the blunders which he was pleased to attribute at Cambridge to me and metaphysics.'

The September number of the Philosophical Magazine accordingly contains an `Additional Note on Conical Refraction, by J. MacCullagh, F.T.C.D.,' which I transcribe:-

`The introductory part of my note which appeared in your last number was written in haste, and I have reason to think it may not be rightly understood. You will therefore allow me to add a few observations that seem to be wanting.

`The principal thing pointed out in the Paper that I published some time ago in the Transactions of the Royal Irish Academy is a very simple relation between the tangent planes of Fresnel's wave surface and the sections of two reciprocal ellipsoids. Now this relation depends upon the axes of the sections, and therefore naturally suggested to me the peculiar cases of circular section in which every diameter is an axis. Thus a new inquiry was opened to my mind. And accordingly, without caring just then to obtain final results, which seemed to me to be an easy matter at any time, I expressed in conversation my intention of returning to the subject of Fresnel's Theory in a supplementary Paper. The design was interrupted, and I was prevented from attending to it again, until I was told that Professor Hamilton had discovered cusps and circles of contact on the wave surface. This reminded me of the cases of circular section, and the details given in my last note were immediately deduced.'

Among Hamilton's papers I find the following note from Mac Cullagh:-

From J. Mac Cullagh to W. R. Hamilton.

`Tuam, September 5, 1833.

`My dear Hamilton, I have not seen the last number of the Phil. Mag. though I ordered it to be sent to me, and I am uneasy to know whether the second Note has been published or not. I made it clearer and more precise by the alteration of a word or two in what regards myself; what relates to you was retained verbatim, and I hope you will find it completely to your satisfaction. If you should think it necessary to say anything yourself, perhaps you would defer doing so until we meet, which may take place in three or four weeks. In the meantime I am anxious to hear from you, as I suppose the Phil. Mag. has gone astray.'

On the back of the above letter is the short-hand draft by Hamilton of his reply:-

`My dear Mac Cullagh, I have just seen your ``Additional Note'' in the Phil. Mag., and have no intention of troubling the editors with any remarks of my own on the subject. They [will] know the rest from some other Papers from you which have not yet been printed.' [12] Then follows a generalisation by Hamilton `of your curious theorem about a refracting hyperboloid:' [13] and he concludes, `On going to the Academy the last day that I saw you, I found they had broken up for the summer, so that I was too late to propose the insertion of any note to my Third Supplement, and the appearance of your own communications in the Magazine seem to make it unnecessary. Believe me, &c.'

In his Introduction, however, to his Third Supplement, printed in part i. of the vol. of the Transactions of the Royal Irish Academy, and dated June 1833, Hamilton had thus put on record the researches of Mac Cullagh in this part of Fresnel's Theory:-

`I am informed that James Mac Cullagh, Esq., F.T.C.D., who published in the last preceding volume of these Transactions a series of elegant Geometrical Illustrations of Fresnel's Theory has, since he heard of the experiments of Professor Lloyd, employed his own geometrical methods to confirm my results respecting the existence of those conoidal cusps and circles on Fresnel's wave from which I had been led to the expectation of conical refraction. And on my lately mentioning to him that I had connected these cusps and circles on Fresnel's wave with circles and cusps of the same kind on a certain other surface discovered by M. Cauchy, by a general theory of reciprocal surfaces, which I stated last year at a general meeting of the Royal Irish Academy, Mr. Mac Cullagh said that he had arrived independently at similar results, and put into my hands a Paper on the subject; which I have not yet been able to examine, but which will I hope be soon presented to the Academy and published in their Transactions.'

To this I may add the acknowledgment, which immediately follows, of the approximation made by Professor Airy to the result arrived at by Hamilton:-

`I ought also to mention that on my writing in last November to Professor Airy, and communicating to him my results respecting the cusps and circles on Fresnel's wave, and my expectation of conical refraction, which had not then been verified, Professor Airy replied that he had long been aware of the existence of the conoidal cusps, which indeed it is surprising that Fresnel did not perceive. Professor Airy, however, had not perceived the existence of the circles of contact, nor had he drawn from either cusps or circles any theory of conical refraction.'

The statements with which the Introduction concludes ought perhaps here to be given to the reader; in continuation with the paragraph last quoted, it proceeds:-

`This latter theory was deduced by my general methods from the hypothesis of transversal vibrations in a luminous ether, which hypothesis seems to have been first proposed by Dr. Young, but to have been independently framed and far more perfectly developed by Fresnel; and from Fresnel's other principle of the existence of three rectangular axes of elasticity within a biaxal crystallized medium. The verification therefore of this theory of conical refraction by the experiments of Professor Lloyd must be considered as affording a new and important probability in favour of Fresnel's views; that is, a new encouragement to reason from those views in combining and predicting appearances.

`The length to which the present Supplement has already extended obliges me to reserve for a future communication many other results deduced by me by my general methods from the principle of the characteristic function; and especially a general theory of the focal lengths and aberrations of optical instruments of revolution.'

In the Third Report of the Proceedings of the British Association for the Advancement of Science, giving the proceedings of the Meeting at Cambridge in June, 1833, but corrected up to the time of printing in 1834, is to be found, at p. 360, a report of Professor Hamilton's oral statement of `Results of a view of a Characteristic Function in Optics.' This embraces some results relating to optical instruments of revolution, as well as Conical Refraction; and it concludes, at p. 369, with a reference to the independent researches of Mac Cullagh and Cauchy. It is followed by a similar report of Professor Lloyd's oral statement of his verifying experiments.

At the close of an article, [14] dated September, 1833, contributed by Hamilton to the November Number of the Dublin University Review for 1833, p. 823, Mr. Mac Cullagh's claim in this matter is also put on record.

These statements of Hamilton with regard to Mac Cullagh's work are all in perfect consistency with one another.

Finally, in the volume of the Transactions of the Royal Irish Academy, part ii., p. 248, Mr. Mac Cullagh put a satisfactory close to his action in the matter, by appending to his Paper entitled Geometrical Propositions applied to the Wave Theory of Light (Read, June 24, 1832), a note dated April 2, 1834, which is here transcribed:-

`The curves of contact on biaxal surfaces and the conical intersections and nodes were lately discovered by Professor Hamilton, who deduced from these properties a theory of conical refraction which has been verified by the experiments of Professor Lloyd. See Transactions Royal Irish Academy, vol. xvii., part i., and the present Paper, Art. 55--58.

`The indeterminate cases of circular section - at least the case of the nodes - had occurred to me long ago; but having neglected to examine the matter attentively, I did not perceive the properties involved in it.'

I have now brought forward or referred to all the facts and contemporary records respecting the question of priority and mutual independence which have come within my cognizance. The reader will see that proceeding by different paths (Hamilton by that of his own Algebraical method, Mac Cullagh by that of Geometry), Hamilton independently completed his theoretical discovery and foresaw the corresponding physical facts: Mac Cullagh, when working independently, advanced far in the right direction, but stopped short of deducing all the connected mathematical properties, and failed to anticipate the physical phenomena to which his theorems might have conducted him.

It may be added with truth that by nothing was Hamilton more distinguished, from the beginning to the end of his scientific career, than by his scrupulous anxiety to award to all labourers in the same fields with himself the shares to which they had a just title in the priority and independence of discovery.


  1. This finally proved to be, quam proxime, the angle ascertained by experiment: see letter to Herschel, infra, p. 634. [The page number here refers to Life of Sir William Rowan Hamilton by Robert Perceval Graves, vol I, Dublin University Press, 1882.]
  2. This I am informed is a mistake.
  3. In the case of external conical refraction.
  4. The correspondence on this subject shows that Hamilton solved it the day after he wrote this letter, viz., January 2.
  5. See Note in the Appendix.
  6. In a letter to another correspondent, dated January 22, 1832, Hamilton writes, `Airy has just answered the letter that I wrote to him from Bayly Farm; but whether the fault was mine or his, he has quite mistaken what I meant.'
  7. See p. 636. [The page number here refers to Life of Sir William Rowan Hamilton by Robert Perceval Graves, vol I, Dublin University Press, 1882.]
  8. The Philosophical Magazine.
  9. Report of third Meeting of the British Association, 1833.
  10. London and Edinburgh Philosophical Magazine, June 1833, p. 420.
  11. The interior cone emerges as a cylinder.
  12. This I suppose to refer to the Paper read by Mac Cullagh before the Royal Irish Academy on the 24th June, 1833. Geometrical Properties, &c., referred to below as subsequently printed in the vol., part ii., of the Royal Irish Academy.
  13. See letter from W. R. Hamilton to Lloyd, dated September 2, 1833.
  14. On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function.


    D.R. Wilkins
    School of Mathematics
    Trinity College, Dublin