429 



comes, for any spherical triangle, in which the rotation round 



R, from r' to r'', is positive, 



(cos R + is, sin Vx) (cos r' + zV sin r') = — cos R" + iw' sin r". (i') 



If p" be the positive pole of the arc rr', or the pole to 

 which the least rotation from r' round r is positive, then the 

 product of the two imaginary units in the first member of 

 this formula (which may be any two such units), is the fol- 

 lowing : 



i^ i^. = — cos rr' + ip' sin rr' ; (m) 



we have also, for the pi'oduct of the same two factors, taken 

 in the opposite order, the expression 



ij^, i^ z= — cos R r' — iff sin r r', (n) 



which diflfers only in the sign of the imaginary part ; and 

 the product of these two products is unity, because, in gene- 

 ral, 

 (w + ix-{-jy-\-kz) {iv—ix—jt/-k^) = iv^-\-x'^-^7/+^^; (o) 



we have, therefore, 



2b H'- k' 2k = 1 5 (P) 



and the products 4 «V and is.' i^. may be said to be reciprocals 

 of each other. 



In general, in virtue of the fundamental equations of de- 

 fiuition, (a), (b), (g), although the distributive character of 

 the multiplication of ordinary algebraic quantities (real or 

 imaginary) extends to the operation of the same name in the 

 theory of quaternions, so that 



q(q' + q'0 = qq'+qq", &c., 

 yet the commutative character is lost, and we cannot gene- 

 rally write for the new as for the old imaginaries, 

 qq' = q'q, 



since we have, for example, Ji = — ij. However, in virtue of 

 the same definitions, it will be found that another important 

 property of the old multiplication is preserved, or extended 



