[Note added during printing.]-The printing of this abstract having been delayed, the Author desires to be permitted to append the following remarks:

If we should make, for abridgment

the formula (i) for any single rotation might be thus written,

And if we then made

i, j, k, being the same three rectangular vectors, or imaginary units, as in the formulæ (A) (B) (C), but x, y, z, x', y', z', , , , being nine real or scalar quantities, we should obtain the same general formula for the transformation of rectangular coordinates (with the same geometrical meanings of the coefficients , , ,) as that which Mr. Cayley has deduced, with a similar view, but by a different process, and has published, with other ``Results respecting Quaternions,'' in the Philosophical Magazine for February, 1845.

The present writer desires to return his sincere acknowledgments to Mr. Cayley for the attention which he has given to the Papers on Quaternions, published in the above-mentioned Magazine: and gladly recognizes his priority, as respects the printing of the formula just now referred to. But while he conceives it to be very likely that Mr. Cayley, who had previously published in the Cambridge Mathematical Journal some elegant researches on the rotation of bodies, may have perceived, not only independently, but at an earlier date than he did himself, the manner of applying quaternions to represent such a rotation; he yet hopes that he may be allowed to mention, that a formula differing only slightly in its notation from the formula (i) of the present abstract, with the corollaries there drawn respecting the composition of successive finite rotations, had been exhibited to his friend and brother Professor, the Rev. Charles Graves, of Trinity College, Dublin, in an early part of the month (October, 1844), which preceded that communication to the Academy, of which an account is given above.

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On Quaternions