410 MISCELLANEOUS PROPOSITIONS. 



378. If in the identity at the top of page 120 we put 



h* 



4t = , we shall obtain 

 x 



hence we infer that 



(7), 



and this may be verified by shewing that this value of <f> (x) 

 satisfies equation (5), 



k n 

 By putting x = h we find that k = - ,^ . 





Assume T = cos 0, which is of course allowable so long as 

 as is not numerically greater than h. 



Then {x >J(x*- A 2 ) }" - A" (cos V^I sin 0} n 



= h* {cos n0 V 1 sin n0] ; 

 , . . h n cos n0 



thUS <P (X) = -l~ > 



^ 

 that is so long as x lies between h and A we have 



r \ h* ( -i x \ i f -i x \ 



9 ( x ) = ^Fi cos n cos * -T = A; cos n [ cos T . 

 r 1 \ hj \ hj 



379. The last result may also be obtained from (4). For 

 put <f> (x) = z ; then (4) gives 



,, . ,. 

 therefore 



77^ - 5^ 

 V ( " - * ) 



