TRANSACTIONS OF THE SECTIONS. J25 



Again, 



<KF) Ei=2 pX 1 +p') { e ~ Fi }' 



sm^ w+^[ ^^n^ si ^ C09 ^ (1+y2)+(1+ ^ E '- F -] ; 



whence 



Thence, denoting the operation 



-l_ W+ ^)_rf_] by R, 



[P^P! . . . QP^ ] E 2 =E 2 + sin x cos x <f>(x), 

 [QPiQ . . . PQP> ] F 2 =F 2 + sin x cos x <f>(x), 

 [QP X Q . . . QPiQ ] E 2 =F 2 + sin x cos x <f>(x), 

 [PiQP! . . . P,QPJ F 2 =E a + sin x cos x <f>(x) ; 

 and in the same way 



[SRS . . . RSR] E 1 ==E 1 + sin x cos x </>(>), 

 [RSR . . . SRS ] F, =F,+ sin x cos a? #0), 

 [RSR . . . RSR] E^F, + sin x cos x cj>(x), 

 [SRS . . . SRS ] F 1 =E 1 + sin x cos x <f>(x). 

 Then, again, 



[w+tf> ^ +4(1+/)^ -l] *— ^fi£y'- - cos *, 



and 



y (W) 2 ^ +4(l+ i r)(l+2r) ^ +l]E t 



l+(l+»^shr\r . ,„ aN 



= //. , ,;. a \ sin x cos a? y (1 +!>')> 



1877. 3 



