318 PROFESSOR KELLAND ON THE THEORETICAL INVESTIGATION OF 



it w, and let 



f g 



d 



r rf /rinjr Y /sin_V 

 J y \ y ) \ z ) 



o Jo \ y J \ * J \ sin r * 

 Now, / d yr~} = TT (lastProb.) 



4a 2 e 



T u Jo \ x I \ sm 



Now, it was proved in my Memoir in the Transactions of the Cambridge Philo- 

 sophical Society, vol. vii. p. 163, that the value of the integral 



r/sin a;\ 2 /sin rm x\ 2 . TT 

 dxt - ) (- _) is 



\ x J \ mart J 2 



2 A 2 6 2 

 v, - gg - = j)g~ a x area ot the aperture left uncovered. 



Now, it ought to be a 2 x same aperture. 

 D = 6 X. 



PBOB. III. Let evw*y thing remain as in the last Problem, except that the aperture 



is an isosceles triangle. 



The vibration at the point whose co-ordinates are p, g, is 



the limits being y-x tan a,y-x tan a ; #=0, x=c cos a : where is the half 

 of the angle included by the equal sides of the triangle. Integrating with respect 

 to y, we get 

 ^ 6a /jf 2 TT / px qx tan a\ 2 ' 



A 6 a /\ f 2 TT / -n P x ytana\ 2 TT / JBJ 



If M, N denote the coefficients respectively of 



cos - ( * B) and sin -, - ( /-B) in this integral, 



M= . -^ ; r-^pr sin A . (p cos aa sin a) 

 2-7T7 STT (p q tan a) D \b ^ 



A6 A6a ,27Tc. ., 



s -o / ^TT sin ^ , ( cos a + q sm a) 



27ry 2 -7T (/> + y tan a) D A6 



27TC , 



_ . ^ , 1 + cos tr-j- (cos a o sm a) 



A b \ba Afi 



2 Try 27rD j y tan a 



2-7TC. 



J . - . 1 cos -^-r- ( p cos a + o sin a; 



\b \ba A o ^ 



^ '2irq 2 TT D p + q tan a 



