80 VECTOR ANALYSIS. 



Now I + gN" 1 !/^ jk} evidently represents a versor having the axis i 

 and the infinitesimal angle of version t/N" 1 . Hence the above ex- 

 ponential represents a versor having the same axis and the angle of 

 version q. If we set qi = o>, the exponential may be written 



e lxw . 



Such an expression therefore represents a versor. The axis and 

 direction of rotation are determined by the direction of a, and the 

 angle of rotation is equal to the magnitude of o>. The value of the 

 versor will not be affected by increasing or diminishing the magnitude 

 of o> by 27T. 



178. If, as in No. 151, 



$ = aaa' + b/3/3' + cyy, 

 the definitions of No. 171 give 



e* = e a aa -f e b /3/3' -f e c yy, 

 cos <& = cos a aa 4- cos b /3/3' + cos c yy, 

 sin <1> = sin a aa + sin b /3/3' + sin c yy. 

 If a, b, c are positive and unequal, we may add, by No. 172, 

 log < = log a aa' -f- log b /3f? -h log c yy. 



179. If, as in No. 153, 



= aaa' + p cos q {ftft' + yy} + p 8mq{yP'-/3y'}, 

 we have by No. 1 73 e * = e <^ % e &{^'+yy'} . e ^ - M. 



But eP* = e a aa' + &f + yy', 



aa' + e b {/3/3' + yy'}, 



Therefore, e * = e a aa ' + ^coscj/^+yy'} + e & sin 

 Hence, if a is positive, 



Since the value of $ is not affected by increasing or diminishing q bi 

 2-7T, the function log $ is many- valued. 



To find the value of cos 3? and sin <3>, let us set 



Then, by No, 175, 



cos $ = cos {aaa'} -f cos I. 



cos {aaa'} I = cos aaa' aa'. 

 Therefore, CQ8 $ = C08 aafl , _ ^ + CQg e 



