11 



direction is perpendicular to both of their's, being obtained 

 from that of jSa, by making it revolve right-handedly through 

 a right angle round a as an axis. These definitions, which 

 are compatible with the formulae (A) (B) (C), and may serve 

 to replace them, will be found suflBcient to prove generally, 

 and perhaps with somewhat greater geometrical clearness than 

 those formula, the distributive and associative properties of 

 quaternion multiplication, which have been already stated to 

 exist. They give easily the following corollaries, which are 

 of very frequent use in this calculus : 



a/3 + jSa = 2aj3i = - 2 AB cos(A, B) ; (a) 



aj3 -i3a = 2ai32 = 27ABsin(A,B); (b) 



A and B denoting here the lengths of the lines a and j3, and 

 (A, B) the angle between them ; while 7 is a vector-unit per- 

 pendicular to their plane, and such that a right-handed rota- 

 tion, equal to the angle (A, B), performed round y, would 

 bring the direction of a to coincide with that of /3. For 

 example, when j3 = a, then B = A, (A, B) = 0, and 



a/3 = ^a = a^ = -^ A^, 

 so that the length A of any vector a, in this theory, may be 

 expressed under the form 



A = \/^^\ (c) 



More generally we have the equation 



ajd-fia- 0, (d) 



when the lines a and /3 are coincident or opposite in direction ; 

 while, on the contrary, the condition for their being at right 

 angles to each other is expressed by the formula 



ali-j- (ia = 0. (e) 



These simple principles suffice to give, in a new way, 

 algebraical solutions of many geometrical problems, of various 

 degrees of difficulty and importance. Thus, if it be required, 

 as an easy instance, to determine the length of the resultant 



