Sir W. Rowan Hamilton o?i Quaternions. 20l 



the surface in which the polygons are inscribed shall be sup- 

 posed, for the present, to be an ellipsoid. Results of the same 

 general character, but with some important modifications, 

 (connected with the ordinary square root of negative unity,) 

 hold good for the inscription of such polygons in other surfaces 

 of the same order, as the writer may afterwards point out. 

 He is a\\'^re, indeed, that the corresponding class of questions, 

 respecting the inscription of pla?ie polygons in conies, has 

 attained sufficient celebrity; and feels that his own acquaint- 

 ance with what has been already done in that department of 

 geometrical science is inferior to the knowledge of its history 

 possessed by several of his contemporaries, for instance, by 

 Professor Davies. He knows also that some of the published 

 methods for inscribing in a circle, or plane conic, a polygon 

 whose sides shall pass through the same number of given 

 points, can be adapted to the case of a polygon formed by arcs 

 of great circles on the surface of a sphere, and inscribed in a 

 spherical conic ; and he has, by quaternions, been conducted 

 to some such methods himself, for the solution of this latter 

 problem. But he acknowledges that he shall feel some little 

 surprise, though perhaps not entitled to do so, if it shall turn 

 out that the results of which he proceeds to give an outline, 

 respecting the inscription of rectilinear hut gauche polygo7is in 

 an ellipsoid, have been wholly (or even partially) anticipated. 

 They have certainly been, in his own case, results of the ap- 

 plication of the quaternion calculus : but whatever geometrical 

 truth has been attained by any one general mathematical me- 

 thod (such as the Quaternions claim to be), may also be found, 

 or at least proved, by any other method equally general. And 

 those who shall take the pains of proving for themselves, by 



he has been engaged, he hopes that he may be allowed to say, — yet rather 

 as requesting a favour than as claiming a right, — that he will be happy if 

 the inventor of the Pluquaternions shall consent to his adopting or rather 

 retaining a wordy namely " biquaternion," which the Rev. Mr. Kirkman 

 has indeed employed, with reference to the octaves of Mr. J. 1\ Graves and 

 Mr. Cayley, but does not appear to want, for any of his own purposes : 

 whereas Sir W. Rowan Hamilton has for years been accustomed to use 

 this word Biquaternion,— though perhaps hitherto without printed publi- 

 cation, — and indeed could not, without sensible inconvenience, have dis. 

 pensed with it, to denote an expression entirely distinct from those octaves, 

 namely one of the form 



where -v/ — 1 is the old and ordinary imaginary of algebra (and is therefore 

 quite distinct from t,j, k), while Q and Q' are abridged symbols for two 

 different qicaternions of the kind w^ix-'rjy-\-kz, introduced into analysis in 

 1843. Biquaternions o^ this sort have repeatedly ybrcerf themselves on the 

 attention of Sir W. R. H., in questions respectmg geometrical impossibility, 

 ideal intersections, imaginary deformations, and the like. 



