Sir W. Rowan Hamilton on Quaternions, 295 



equations (120.), respectively equal, by (86.), (91.), to those 

 which have been denoted above hy h and h'. In like manner, 

 by (110.), 



^^'=,-^=V-0; ez:^ = V^,=V-'0. (121.) 



And because, by (107.), / has a scalar ratio to x, and x' has 

 a scalar ratio to i, we may infer, from (118.), (119.), the ex- 

 istence of the two following other scalar ratios : 



ttlh^Y-'O; htl^' =Y-'0. . . (122.) 



Finally we may observe that, by (120.), (121.), there exist 

 scalar ratios between certain others also of the foregoing vec- 

 tor-differences, and especially the following : 



^-^::^=v-^o; ^nV-v-'o. . . . (123.) 

 p-h p-h 



66. Proceeding now to consider the geometrical significa- 

 tion of the equations in the last article, we see first, from the 

 equations (117.)} that the four new points, l^ m^, l/, m/, are 

 all situated upon the surface of that mean sphere, which is 

 described on the mean axis of the ellipsoid as a diameter; 

 because the equation of that mean sphere has been already 

 seen to be 



p2 + /;2=o* equation (100.), article 58; 



which may also be thus written, by the principles and nota- 

 tions of the calculus of quaternions : 



Tp = b • (124.) 



From the relations (122.) it follows that the two chords l^m/ 

 and l/Mp of this mean sphere, both pass through the point h, 

 of which the vector ^^ is assigned by the foimula (116.) ; for 



* Thh form of the equation of the sphere was published in the Philo- 

 sophical Magazine for July 1846 ; and it is an immediate and a very easy 

 consequence of that fundamental formula of the whole theory of Quater- 

 nions, namely 



which was communicated under a slightly more developed form, to the 

 Royal Irish Academy, on the 13th of November 1843. (See Phil. Mag. for 

 July 1844.) 



It may perhaps be thought not unworthy of curious notice hereafter, that 

 after the publication of this form of the equation of the sphere, there should 

 have been found in England, and in 1846, a person with any mathematical- 

 character to lose, who could profess publicly his inability to distinguish the 

 method of quaternions horn that of couples ', and who could thus confound 

 the system of the present writer with tliose of Argand and of Fran9ais, of 

 Mourey and of Warren. 



