478 ON CERTAIN APPROXIMATE FORMULAE FOR [59 



/ 7 \ ** 



It will be noticed that the term involving i -J-JT) has disappeared by this 

 division. 



Now make m 2, and this formula becomes 





( i r / ^ M \ _ __^_ _ _ -] i 



-M - (a-/3) 2 2/(i-l) I ,,) tany + n-1 n + 4 (secy) 2 -TO -1 n + 3 L 

 ^ 24 \_ \ua<f>/ JJ 



Divide throughout by g, and put 1 = 2, then, from before, 



- =ir < - W [ 



Similarly, divide throughout by g, and put 1=1, then 



\l -~ - (a-- ft)' 2 (-^T) tany + (n + 4)(secy) 2 -n + 3 1 



[ 24 \ua<p/ JJ 



Lastly, let 



= u l (sec ^>)'" tan <j> =f(<f>) tan ^> suppose, 



so that 



/(<) = < (sec <)'; 



then F'(^) =f ((f>) tan ^ +/(<) (sec </.) 2 , 



and .F" (<) =/" (^>) tan <^ + 2/' (</>) (sec <^) 2 + 2/ (<) (sec ^)) 2 tan ^. 



Hence f " ,F (^) d<f> = (a- 0) {F (y) + ^(a- /8) 2 F" (y) 1 approximately, 



J^ Z4 



= (-) {/(y) tan y + ^ (a - ^) 2 [/"(y) tany + 2/'(y) (secy) 2 + 2/(y) (secy) 2 tany]| 



also 



fa \ 1 I 



approximately ; 



