Mr Sharpe, On the Reflection of Sound at a Pai^aboloid. 125 



We at once get 



f smy 3 dy = ir(^\ . sin |= -07132, &c, 



by De Morgan's Differential Calcidus, p. 590. 

 By means of a figure of the integral 



/, 



^v~% cos vdv (which is equal to I cos a? dx) 



o Jo 



it can be shewn that 



r ("-/ajf* 



cos x s doc = I cos a? dx 



Jo 



/•OO 



— something which is less than / sina^dx, 



Jo 



from which we are able to prove that I cos X s dx is positive. 



Jo 

 We must therefore reject the possible value (3) above, and we 

 get 



J o cos fdy = \ r(|). cos I . 



Next in (76) putting n = l, we shall find that the function of 

 7r on the right of (76) will take three possible values, which are 



(1) cos - — i sin ■=■ , (2) cos it — i sin it, (3) cos - + i sin - . 



O O O O 



It is easily shewn that the first is the value to be chosen, and 

 we get 



/ oS iny%=|r(|). S m|. 

 Finally, ^(D'xirQ.cosf 



*Tr,/*-!(!)\r(i),co.J 



.(77). 



29. We will next get the values of v which make V 1 and 

 dV 1 /dv maxima. We will begin with dV/dv as the most important 

 for our purpose, and first consider equation (12) which is (dropping 

 the dashes) 



d?V dV . ,, Tr A 

 V d^ + Tv + ^-^ V =°' 



VOL. X. PT. III. 10 



