110 PRACTICAL MATHEMATICS 



When x = 0, sec x and its successive differential coefficients 

 become 1, 0, 1, 0, 5, 0, 61, and therefore there are no odd powers 

 of x in the expansion. 



67. There are some functions which can be expanded without 

 the aid of Maclaurin's Theorem, and this remark applies more to 

 the inverse trigonometrical functions. 



To find an expansion for sin" 1 x. 



y = sin- 1 x = A + K^x + A 2 # 2 + A 3 # 3 + A 4 a? 4 + A 5 # 5 + . . . 

 and when x = 0, y = 0, and therefore A = 



3A 3 a: 2 



But -7=== or (1 x 2 ) 2 can be expanded with the aid of the 

 V 1 x 2 



Binomial Theorem and 



Equating coefficients of like powers of x, it should be noticed 

 that there are no terms in the second expansion involving odd 

 powers of x, and therefore the coefficients of these terms in the 

 first expansion must all be zero. 



35 



128 



35 

 1152 



2-4-6 7 ' 2-4-6-8 9 + ' ' ' 



68. If we have a simple relation between the first and second 

 differential coefficients, and this is combined with a knowledge 

 of some of the initial conditions, we are enabled to express y as 



