120 EXAMPLES OF 



By adding the expansions we have found above we obtain 

 . 



+ 



and by what we have just shewn the series on the right hand 

 extends to - + 1 terms if n be even, and to - - terms if n 



a A 



be odd, so that the remaining terms in the two expansions 

 must disappear ; that is, the terms arising from one expan- 

 sion are cancelled by similar terms arising from the other. 

 In the same manner as we deduced the expansion of y* from 



sc 

 the equation y = n-\ _ *F\ we mav deduce the expansion 



of any other function of y ; for example take log y. Thus 

 logy = l 



where after the differentiations are performed we must put 

 x fx\~ . 3 fxv .4.5 fx\ 6 5.6.7 



/v 



- for z. Therefore 



m 



O ' V 9 / ' 9 V 9 / '9 9 I o / 'o > illol "! 



Z \4/ ^ VZ/ Z . O 





Let a? 8 = 4f, and we shall obtain 



The expansions which this example has furnished are of 

 some importance in mathematics. 



5. If x=y&, expand sin (a +y) in powers of x. 



We have given y = xe~ v . Suppose then y = z + xe"", so that 



) = e -*, and/(y) = sin (a + y). 

 The general term given by Lagrange's Theorem is 



