SECT. IV.] VALUES OF THE INTEGRALS. 415 



and = \dy cos y* \dy siny 2 , 



or 



It remains only to determine, by means of the equations (a) 

 and (6), the values of the two integrals 



I dy cos y z cos 2by and | dy sin y* sin 2by. 

 They can be expressed thus : 



A = I dy cos y* cos 2by = h sin 6 2 + g cos 5 2 , 



B = I dy sin i/ 2 cos 26^ = h cos 6 2 - ^ sin b 2 ; 

 whence we conclude 



writing sin ^ , or cos ^ instead of i/ - , we have 



= ^ s ing4-|) .................. (1) 



and I dpsmtft cospz= -ILsmt 7 ^^-}.., ,..(2) 



/- 4 kt) 



408. The proposition expressed by equation (B) Article 404, 

 or by equation (E) Article 361, which has served to discover the 

 integral (8) and the preceding integrals, is evidently applicable to 

 a very great number of variables. In fact, in the general equation 





J /+*&amp;gt; -+QO 



or / 0) = 9- / d P\ d* cos (px - p*)f (a), 



A- V&amp;lt; . OO J ~ 3D 



