429 



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

 K, from r' to r", is positive, 



(cos R + /b sin Ft) (cos r' + ?«■ sin r') = — cos r" + V/ 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^ ia- = — cos rr'+ ?p' sin rr'; (m) 



we have also, for the product of the same two factors, taken 

 in the opposite order, the expression 



?B' ?j = — cos R r' — if'/ sin r r', (n) 



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

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

 ral, 

 (tv{-ix-^Jy + /cs:) {tv—ix—Jtj-kz) = w'^-\-x'^-{-}/-\-z'^; (o) 



we have, therefore, 



«eV. i^'i^ =1, (p) 



and the products i^ is.' and ?b' ?» may be said to be reciprocals 

 of each other. 



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

 finition, (a), (b), (c), 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") = 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,_;« = — ij. However, in virtue of 

 the same definitions, it will be found that another important 

 property of the old multiplication is preserved, or extended 



