1889.] A. Mn]ihoTpn.dhya.j—ElUpiic Functions and Mean Values. 205 

 (a-6)M = 7raZ&-l7rZ<S 



2 di 



+ 



(a^ cosS i + sin" 

 Now, we have 



(a2 cos^ ^ + il? sm2 # "'J^ 



where the eccentricity of the ellipse is given by 



But we have generally 



/ 0 ~ ('^ SUV 



^ sill 9 cos 



Tlicroforc, we get 



r dt 



J„ (l-r^sin 



= I (l-'-^sin^e, . 



I ^„ (1 - <'2siu2 6l^2 ) 



We have also 



cos2 f + sinS (] _ e2 sin2 i)^ 



= aHl-e^) f^^ (l-e2siu!Sf)^rf|. 



(a^cosH + bHin^ i)i d^=aj'^^^ (l-e^am^i)i di, 

 where, as before, 



Hence, 



(a -h)M= 7ra% - ^ + 5 1^ ^ ~ '"'^ ^ 



* See Dr. Scliloemilcli'a Thcoiie lies IiMgrales et des Fonclions ElllpHques, par 

 Dr. Graiiidorge, (1873), p. 14. 



