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unknown, or in some sense transcendental as compared with 
+, so that it would be impossible to express a term affected 
with the sign +? by means of any combination of terms affected 
with the sign +. With this understanding we might call 
terms affected with +* imaginary, and the importance of the 
distinction between reals and imaginaries consists in this: that 
when we have an equation, u = 0, involving quantities of 
both kinds, we may put the real and imaginary parts of u 
respectively equal to 0. 
The following example will show the advantage of employ- 
ing such asymbol. If we put + in place of + @ in the 
development of ¢*, we find 
+3 ; 
e * =cosp +? sing (1) 
where cog and ging respectively stand for the series 
¢* 
iat ©: 
Te 
ae - 
es ee Re 
multiplying equation (1) by the similar one 
3, 
et — cog’ +* ging’ 
we have 
peice 
pene ae cosp cos’ + sing sing’ +* [cos sing’ + sing cosp’ |. 
But again, from (1) we have 
+3 , ‘ 
e PF = cos(p + o’) +4 sin(p + o’). 
And since we are entitled to compare the real and imaginary 
parts in the two last equations, we conclude that 
cos(p + ’) = cosp cosd’ + sing sing’ 
sin(p + ¢’) = cosd sing’ + sing cos’. 
