476 ON CERTAIN APPROXIMATE FORMULAE FOR [59 



Hence 



F"($) = (n+ l)[(w + 1) (sec <) B+1 (tan <)' + (sec <) n+3 ] 



= (n + 1 ) [w + 2 (sec <f>) n+3 - w+1 (sec <) n+1 ] ; 

 and therefore, 



f " (sec <f>) n+1 d<t> = (a- ft) (sec y)" +1 {l + ^ - 1 (a - /3) 2 [JT+2 (sec y) 2 - 

 J ft I 



to the 4th order inclusive. 

 Hence 



- - - 2 ~ (a - /3) 2 [^T2 (sec y) 2 - 



L iy 



which gives q when p is known. 



In the next place, let F (<f>) = u l (sec <f>) m . 

 Hence 



= lu l ~ l (sec <fy m + mu l (sec <) m tan 



or F'(<j>) = F(<f>) |~~ u n (sec </>) n+1 + m tan <~| ; 



I V7 I 



and 



- u n (sec 4>) n+1 + m tan 



9 



or 



FZ-7-n <7w 1?1 ~\ 



\ u n - 1 ^ (sec ^,)" +1 +-(n+l)u n (sec ^>) n+l tan <^ + m (sec <)' , 



' i/ T^ \J ' 



T.2/2 u m -\ 



^ T u m (sec <) 2n+2 + 2 ^ it" (sec <) n+1 tan ^ + m 2 (sec <^)) 2 - m 2 



t/ t/ I 



tP7 ^-7 n 



~ w 2 " (sec ^) w+2 + - (n + 1 ) w" (sec <^.)" +1 tan < + m (sec ^) 2 

 i/ i/ I 





