ON SOME RESULTS OBTAINED BY THE QUATERNION ANALYSIS RESPECTING

THE INSCRIPTION OF “GAUCHE” POLYGONS IN SURFACES OF THE

SECOND ORDER

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 4 (1850), pp. 380–387.)

Edited by David R. Wilkins On some Results obtained by the Quaternion Analysis respecting the inscription of “gauche” Polygons in Surfaces of the Second Order. By Sir William R. Hamilton.

Communicated June 25, 1849.

[Proceedings of the Royal Irish Academy, vol. 4 (1850), pp. 380–387.]

Sir William Rowan Hamilton communicated to the Academy some results, obtained by the quaternion analysis, respecting the inscription of gauche polygons in surfaces of the second order.

If it be required to inscribe a rectilinear polygon p, p_{1}, p_{2}, … p_{n−1} in such a surface, under the conditions that its n successive sides pp_{1}, p_{1}p_{2}, … p_{n−1}p, shall pass respectively through n given points, a_{1}, a_{2}, … a_{n}, the analysis of Sir W. R. H. conducts to one, or to two real lines, as containing the first corner p, according as the number n of sides is odd or even: while, in the latter of these two cases, the two real right lines thus found are reciprocal polars of each other, with reference to the surface in which the polygon is to be inscribed. Thus, for the inscription of a plane triangle, or of a gauche pentagon, heptagon, &c., in a surface of the second order, where three, five, seven, &c, points are given upon its sides, a single right line is found, which may or may not intersect the surface; and the problem of inscription admits generally of two real or of two imaginary solutions. But for the inscription of a gauche quadrilateral, hexagon, octagon, &c., when four, six, eight, &c. points are given on its successive sides, two real right lines are found, which (as above stated) are polars of each other; and therefore, if the surface be an ellipsoid, or a hyperboloid of two sheets, the problem admits generally of two real and of two imaginary solutions: while if the surface be a hyperboloid of one sheet, the four solutions are then, in general, together real, or together imaginary.

When a gauche polygon, or polygon with 2m + 1 sides, is to be inscribed in an ellipsoid or in a double-sheeted hyperboloid, and when the single straight line, found as above, lies wholly outside the surface, so as to give two imaginary solutions of the problem as at first proposed, this line is still not useless geometrically; for its reciprocal polar intersects the surface in two real points, of which each is the first corner of an inscribed decagon, or polygon with 4m + 2 sides, whose 2m + 1 pairs of opposite sides intersect each other respectively in 2m + 1 given points, a_{1}, a_{2}, … a_{2m+1}. Thus when, in the well-known problem of inscribing a triangle in a plane conic, whose sides shall pass through three given points, the known rectilinear locus of the first corner is found to have no real intersection with the conic, so that the problem, as usually viewed, admits of no real solution, and that the inscription of the triangle becomes geometrically impossible; we have only to conceive an ellipsoid, or a doublesheeted hyperboloid, to be so constructed as to contain the given conic upon its surface; and then to take, with respect to this surface, the polar of this known right line, in order to obtain two real or geometrically possible solutions of another problem, not less interesting: since this rectilinear polar will cut the surface in two real points, of which each is the first corner of an inscribed gauche hexagon whose opposite sides intersect each other in the three points proposed. (It may be noticed that the three diagonals of this gauche hexagon, or the three right lines joining each corner to the opposite one, intersect each other in one common point, namely, in the pole of the given plane.)

If we seek to inscribe a polygon of 4m sides in a surface of the second order, under the condition that its opposite sides shall intersect respectively in 2m given points, the quaternion analysis conducts generally to two polar right lines, as loci of the first corner, which lines are the same with those that would be otherwise found as loci of the first corner of an inscribed polygon of 2m sides, passing respectively through the 2m given points. Thus, in general, the polygon of 4m sides, found as above, is merely the polygon of 2m sides, with each side twice traversed by the motion of a point along its perimeter. But if a certain condition be satisfied, by a certain arrangement of the 2m given points in space; namely, if the last point a_{2m} be on that real right line which is the locus of the first corner of a real or imaginary inscribed polygon of 2m − 1 sides, which pass respectively through the first 2m − 1 given points a_{1}, … a_{2m−1}; then the inscribed polygon of 4m distinct sides becomes not only possible but indeterminate, its first corner being in this case allowed to take any position on the surface. For example, if the two triangles p′p′_{1}p_{2}′, p′′p′′_{1}p_{2}′′ be inscribed in a conic, so that the corresponding sides p′p′_{1} and p′′p′′_{1} intersect each other in a_{1}; p′_{1}p′_{2} and p′′_{1}p′′_{2} in a_{2}; and p′_{2}p′ and p′′_{2}p′′ in a_{3}; and if we take a fourth point a_{4} on the right line p′p′′, and conceive any surface of the second order constructed so as to contain the given conic; then any point p, on this surface, is fit to be the first corner of a plane or gauche octagon, p p_{1} … p_{7}, inscribed in the surface, so that the first and fifth sides pp_{1}, p_{4}p_{5} shall intersect in a_{1}; the second and sixth sides in a_{2}; the third and seventh sides in a_{3}; and the fourth and eighth in a_{4}. And generally if 2m given points be points of intersection of opposite sides of any one inscribed polygon of 4m sides, the same 2m points are then fit to be intersections of opposite sides of infinitely many other inscribed polygons, plane or gauche, of 4m sides. A very elementary example is furnished by an inscribed plane quadrilateral, of which the two points of meeting of opposite sides are well known to be conjugate, relatively to the conic or to the surface, and are adapted to be the points of meeting of opposite sides of infinitely many other inscribed quadrilaterals.

When all the sides but one, of an inscribed gauche polygon, pass through given points, the remaining side may be said generally to be doubly tangent to a real or imaginary surface of the fourth order, which separates itself into two real or imaginary surfaces of the second order, having real or imaginary double contact with the original surface of the second order, and with each other. If the original surface be an ellipsoid (e), and if the number of sides of the inscribed polygon, pp_{1}, … p_{2m}, be odd, = 2m + 1, so that the number of fixed points a_{1}, … a_{2m} is even, = 2m, then the two surfaces enveloped by the last side p_{2m}p are a real inscribed ellipsoid (e′), and a real exscribed hyperboloid of two sheets (e′′); and these three surfaces (e) (e′) (e′′) touch each other at the two real points b, b′, which are the first corners of two inscribed polygons bb_{1}, … b_{2m−1} and b′b′_{1}, … b′_{2m−1}, whose 2m sides pass respectively through the 2m given points (a). If these three surfaces of the second order be cut by any three planes parallel to either of the two common tangent planes at b and b′, the sections are three similar and similarly placed ellipses; thus b and b′ are two of the four umbilics of the ellipsoid (e′), and also of the hyperboloid (e′′), when the original surface e is a sphere. The closing chords p_{2m}p touch a series of real curves (c′) on (e′), and also another series of real curves (c′′) on (e′′), which curves are the arêtes de rebroussement of two series of developable surfaces, (d′) and (d′′), into which latter surfaces the closing chords arrange themselves; but these two sets of developable surfaces are not generally rectangular to each other, and consequently the closing chords themselves are not generally perpendicular to any one common surface. However, when (e) is a sphere, the developable surfaces cut it in two series of curves, (f′), (f′′), which everywhere cross each other at right angles; and generally at any point p on (e), the tangents to the two curves (f′) and (f′′) are parallel to two conjugate semidiameters.

The centres of the three surfaces of the second order are placed on one straight line; and every closing chord p_{2m}p is cut harmonically at the points where it touches the two surfaces (e′), (e′′), or the two curves (c′), (c′′), which are the arêtes of the two developable surfaces (d′), (d′′), passing through that chord p_{2m}p. In a certain class of cases the three surfaces (e), (e′), (e′′) are all of revolution, round one common axis; and when this happens, the curves (c′), (c′′), (f′), (f′′) are certain spires upon these surfaces, having this common character, that for any one such spire equal rotations round the axis give equal anharmonic ratios: or that, more fully, if on a spire (c′), for example, there be taken two pairs of points c′_{1}, c′_{2} and c′_{3}, c′_{4}, and if these be projected on the axis bb′ in points g′_{1}, g′_{2} and g′_{3}, g′_{4}, then the rectangle bg′_{1} . g′_{2}b′ will be to the rectangle bg′_{2} . g′_{1}b′, as bg′_{3} . g′_{4}b′, to bg′_{4} . g′_{3}b′, if the dihedral angle c′_{1}bb′c′_{2} be equal to the dihedral angle c′_{3}bb′c′_{4}. In another extensive class of cases the hyperboloid or two sheets (e′′) reduces itself to a pair of planes, touching the given ellipsoid (e) in the points b and b′; and then the prolongations of the closing chords, p_{2m}p, all meet the right line of intersection of these two tangent planes: or the inscribed ellipsoid (e′) may reduce itself to the right line bb′, which is, in that case, crossed by all the closing chords. For example, if the first four sides of an inscribed gauche pentagon pass respectively through four given points, which are all in one common plane, then the fifth side of the pentagon intersects a fixed right line in that plane.

An example of imaginary envelopes is suggested by the problem of inscribing a gauche quadrilateral, hexagon, or polygon of 2m sides in an ellipsoid, all the sides but the last being obliged to pass through fixed points. In this problem the last side may be said to touch two imaginary surfaces of the second order, which intersect each other in two real or imaginary conics, situated in two real planes; and when these two conics are real, they touch the original ellipsoid in two real and common points, which are the two positions of the first corner of an inscribed polygon, whose sides pass through the 2m − 1 fixed points. Every rectilinear tangent to either conic is a closing chord p_{2m−1}p; but no position of that closing chord, which is not thus a tangent to one or other of these conics, is intersected anywhere by any infinitely near chord of the system. These results were illustrated by an example, in which there were three given points; one conic was the known envelope of the fourth side of a plane inscribed quadrilateral; and this was found to be the ellipse de gorge of a certain single-sheeted hyperboloid, a certain section of which hyperboloid, by a plane perpendicular to the plane of the ellipse, gave the hyperbola which was, in this example, the other real conic, and was thus situated in a plane perpendicular to the plane of the ellipse. And to illustrate the imaginary character of the enveloped surfaces, or the general non-intersection (in this example) of infinitely near positions of the closing chords in space, one such chord was selected, and it was shown that all the infinitely near chords, which made with this chord equal and infinitesimal angles, were generatrices (of one common system) of an infinitely thin and single-sheeted hyperboloid.

Conceive that any rectilinear polygon of n sides, bb_{1}, … b_{n−1}, has been inscribed in any surface of the second order, and that n points a_{1}, … a_{n} have been assumed on its n sides, bb_{1}, … b_{n−1}b. Take then at pleasure any point p upon the same surface, and draw the chords pa_{1}p_{1}, … p_{n−1}a_{n}p_{n}, passing respectively through the n points (a). Again begin with p_{n}, and draw, through the same n points (a), n other successive chords, p_{n}a_{1}p_{n+1}, … p_{2n−1}a_{n}p_{2n}. Again, draw the n chords, p_{2n}a_{1}p_{2n+1}, … p_{3n−1}a_{n}p_{3n}. Draw tangent planes at p_{n} and p_{2n}, meeting the two new chords pp_{2n} and p_{n}p_{3n} in points r, r′; and draw any rectilinear tangent bc at b. Then one or other of the two following theorems will hold good, according as n is an odd or even number. When n is odd, the three points brr′ will be situated in one straight line. When n is even, the three pyramids which have bc for a common edge, and have for their edges respectively opposite thereto the three chords pp_{2n}, p_{2n}p_{n}, p_{n}p_{3n}, being divided respectively by the squares of those three chords, and multiplied by the squares of the three respectively parallel semidiameters of the surface, and being also taken with algebraic signs which it is easy to determine, have their sum equal to zero. Both theorems conduct to a form of Poncelet’s construction (the present writer’s knowledge of which is derived chiefly from the valuable work on Conic Sections, by the Rev. George Salmon, F. T. C. D.), when applied to the problem of inscribing a polygon in a plane conic: and the second theorem may easily be stated generally under a graphic intesd of a metric form.

The analysis by which these results, and others connected with them, have been obtained, appears to the author to be sufficiently simple, as least if regard be had to the novelty and difficulty of some of the questions to which it has been thus applied; but he conceives that it would occupy too large a space in the Proceedings, if he were to give any account of it in them: and he proposes, with the permission of the Council, to publish his calculations as an appendage to his Second Series of Researches respecting Quaternions, in the Transactions of the Academy. He would only further observe, on the present occasion, that he has made, in these investigations, a frequent use of expressions of the form q + √(−1)q′, where √(−1) is the ordinary imaginary of the older algebra, while q and q′ are two different quaternions, of the kind introduced by him into analysis in 1843, involving the three new imaginaries i, j, k, for which the fundamental formula,

i^{2} = j^{2} = k^{2} = ijk = −1,

holds good. (See the Proceedings of November 13th, 1843.)

And Sir W. R. Hamilton thinks that the name “biquaternion,” which he has been for a considerable time accustomed to apply, in his own researches, to an expression of this form q + √(−1)q′, is a designation more appropriate to such expressions than to the entirely different (but very interesting) octonomials of Messrs. J. T. Graves and Arthur Cayley, to which Octaves the Rev. Mr. Kirkman, in his paper on Pluquaternions, has suggested (though with all courtesy towards the present author), that the name of biquaternion might be applied.