Motion of a Rigid System about a Fixed Point, 431 



ciples of interpretation alluded to, which appear to me to 

 possess considerable advantages, at least for some purposes. 



Returning to the suppositions made at the beginning of this 

 paper, let us conceive two systems of rectangular axes, having 

 their common origin at the centre ; one system being fixed in 

 space, and the other invariably connected with the moveable 

 sphere ; the axes being, moreover, so arranged, that the posi- 

 tive axis of Z shall meet the sphere in the positive pole of the 

 rotation from X to Y. 



Now let ABC be any triangle on the fixed sphere, and let 



AB=id, BC=^fi', AC=ld". 



Also let 



/, m, n ; Z', w', w' ; Z", w", «" 



be respectively the coordinates of the positive poles of the ro- 

 tations AB, BC, AC. Then if 



17= cos - + sin- {il-\-jm + kn), 



and q', q" denote quaternions similarly composed of accented 

 letters, we have, by the principles of the application of qua- 

 ternions to spherical triangles, 



\"=Q'q (1-) 



On the other hand, we have seen that the rotation 2AC( = i9") 

 is the effect of the successive rotations 2AB, 2BC; we have 

 therefore the following equations derived from (1.), and de- 

 fining the rotation which is equivalent to two given successive 

 rotations: — (put, for shortness, 



a:= cos -, X=/tan-, ju,=mtan-, v=wtan-, 



with similar substitutions for the accented letters; then) 



W = M'( 1 - AX' - ^^^ - vv') 



Fx"=Ayt'(x + V+/;t'v-jav') 



^V" = ^-^V + 1^' + "'^ - "^O 



Fv" = /://( V + v' + aV - A/a') . 



Formulae equivalent to these were obtained by Mr. Cayley 

 as analytical results of M. Rodrigues' formulae for the trans- 

 formation of coordinates, and published in the Cambridge 

 Mathematical Journal in IS^S (vol. iii. p. 226). Their geo- 

 metrica significance, however, was not apparent before the 

 d scovery of quaternions. The possibility of representing the 

 effect of successive rotations of a rigid system by a quaternion 



