430 



to the new, namely, that which may be called the associative 

 chtiracter of the o})eration, and which may have for its type 

 the formula 



Q. q'q". «'", Q^^'r: qq'. q"q'"q^^; 

 thus we have, generally, 



Q. q'q" = qq'. q", (q) 



Q. q'«"q'" = qq'. q"q'" = qq'q". q'", (q') 



and so on for any number of factors ; the notation q q'q" 

 being employed to express that one determined quaternion, 

 which, in virtue of the theorem (q), is obtained, whether we 

 first multiply q" as a multiplicand by q' as a multiplier, and 

 then multiply the product q'q" as a multiplicand by q as a 

 multiplier; or multiply first q' by Q, and then q" by qq'. 

 With the help of this principle, we might easily prove the 

 equation (p), by observing that its first member — ?r ?'»/ i^ =. 



In the same manner it is seen at once that 



in k' . ?B' «B" . ?a" is.'" . • . . ^■„(n-l) «'e = (— 1)", (p') 



whatever n points upon the spheric surface may be denoted 

 by R, k', r", r'", . . . r'"-'): and by combining this principle 

 with that expressed by (m), it is not difficult to prove that 

 for any spherical polygon, r r' . . . r'"-'), the following for- 

 mula holds good : 



(cos R + is. sin r) (cos r' + ?»' sin r') (cos r" + '»// sin r") 



. . . (cos R^«-') + y„_,) sinR'"-') = (—])", (r) 



which includes the theorem (i') for the case of a spherical 

 triangle, and in which the arrangement of the m points may 

 be supposed, for simplicity, to be such that the rotations 

 round R from r' to r", round r' from r" to r'", and so on, 

 are all positive, and each less than two right angles, though 

 it is easy to interpret the expression so as to include also the 

 cases where any or all of these conditions are violated. 

 When the polygon becomes infinitely small, and therefore 



