204 A. Mukhopadliyay — TSlliptic Ftmctions and Mean Vahies. [No. 2, 



+ la%^f~^ it .. 



J {a^ sin^ 0 -j- 6^ cos^ <?>)^ 



To effect further reductions, we observe that generally 



j (a^ sin^ 61+62 cos2 d 6 = j (a^ cos^ 6 + ^2 sinS ^1)1 j 

 for, putting 



so that, when 



"2' 



61' = 0, 



,l9=.-dff, 



we get, by substitution, 



(o2 sin2 e + 62 cos2 ey d9 = ^ (a2 cos2 6' + 62 sin'^ 6')^ d 9', 



= 1 (a2cos2 61 + 62sin2 5)2 (^^, 



as the variable is of no consequence in a definite integral. By a similar 

 reasoning, we have 



r i ^ rl 



J „ (a2 sin2 'I' + 62 co.s2 f>)* J ^ (n^ cos^ </> + 62 sin2 <l>y^ 

 and in general 



j'' ^ (a2 sin2 9!, + 62 cos2 0)''' (| 0 = J^|' (a2 co.s2 ■/> + 62 sin2 ^>)'* <7 ^, 

 which is a particalar case of the more general formula 

 I f{sinx) dx=\ f{oosx)dx. 



Therefore, we have 



