Prof. Baker, On a set of transformations of rectangular axes 153 



Further a succession of any number of rotations about axes 

 through the vertex (0, 0, 0, 1), namely a transformation repre- 

 sentable in the form 



is, in virtue of the commutative property of any ^^-transformation 

 with any ^-transformation, capable of being regarded as 



{P,P,P,...){Q,Q,Q,...) 



wherein the first factor is a 2?- transformation, and the second factor 

 a g'- transformation. The laws of composition of the rotations are 

 then precisely the same as those of the associated p-transforma- 

 tions — and the representative point for a composite transformation 

 can be obtained by the composition of the ^-transformations. 



This suffices to reduce the theorem we have stated to the 

 theorem that the representative points of the sixteen transforma- 

 tions referred to are those of the transformations 



(ij, k,co)P{i,j, k,oj), 



where P is any ^j-transformation. For we can pass from any figure 

 on a sphere to any congruent figure by an appropriate rotation PQ. 

 But it is at once obvious that the representative point of a 

 composite transformation PP' may be obtained by forming the 

 product of the symbols associated therewith 



{ai + bj + ck + dcjo) {a'i + b'j + c'k + d'oj) 



by means of the multiplication rules for i, j, k, cu given above. In 

 fact this product gives 



{ad'+ a'd -f he'- b'c) i + {bd'+ b'd + ca'- c'a)j 



+ {cd'+ c'd + ab'- a'b) k 4- {dd'- aa'- bb'- cc') co, 



or say Ai + Bj + Ck + Deo, 



while the product of the two matrices P, P' is at once verified to 

 be the same function oi A, B, C, D as is P of a, b, c, d. 



The theorem stated is thus proved. 



Remark. Any rotation is thus associated with a quaternion, 

 whose vector coefficients give the direction of the axis of the 

 rotation, the amplitude of this being twice the angle whose cotangent 

 is d {a^ + b^ + c^)"*. In particular the symbol i is associated with 

 a rotation about the axis of x of amplitude tt. 



If two rotations of amplitudes 6, 6' about axes {I, m, n), {I,' m',n') 

 be equivalent to a rotation of amplitude cf) about an axis (A, fx, v), 

 we have such equations as 



A sin 1(f) = I sin Id cos |^'+ I' sin |^'cos 1$ + {mn'— m'n) sin \d sin \d' , 

 cos \(f> = cos \d cos \d'— sin |^ sin \d' {W+ mm'-{- nn'). 



