526 PROFESSOU KELLAND ON THE VIBRATIONS OF 



= ? I IH^- y4 sin ft - , f . . ,- dpdrdcf) sinp 

 JJJ r / 8 r 3 sm 3 < 



fromp = to p=fr sin 



= p if ' \ r 2 f -2 + 2 cos (/r sin (p) + 2fr sin sin (/r sin ft) 



*J \J J T Sin Q) i. 



f 2 r 2 sin 2 cos (Jr sin 0) \ dr d(f> 



This cannot be integrated with respect to ft by any known process ; but the 

 form of the result (supposed to vanish at the upper limit) is 



Hence M 



e, 

 the integrals for x, being all negative 



SimUarly 7 + 7 = 



M F 

 Another equation + -r = must be employed in reducing the equations in 



a symmetrical medium. But, although I am satisfied of the truth of the equa- 

 tion, I am not prepared to establish it by direct reasoning. 



The application which I propose to make of the equations now established, 

 is, to determine the phase and intensity of the reflected vibration in the case 

 where there is no proper transmitted one. 



-2 -2 



D --D F 



For the sake of brevity let us write a for 2 , and b for 2 __ ', s for ^ ; 



M ~~M~ 



the medium not being necessarily symmetrical. 

 Our equations of motion thus become (#=0). 

 (I-R)sin^ + R a = T a ................ . . (1) 



(I + R)costf> + R y = T,, .................. (2) 



I'-R' + R, = T* ............. ....... (3) 



,_ / dl dR\s'm(f) dEy I d^ y 1 



a R a + b T a + I -j -- -j I -- r -- -j- 2 - + a ^ - = .... (4) 

 \dx ax ) e dy e dy e \ ' 



R AT f dl dR \<> os( t> dR* 1 fl rf T,l 



a K y + o L y + ( -- f- -j I - '- -- j + a = . . . . (5) 

 \dx dx ) e dy e dye 



i'-4^V = ........ ..... .. (6) 



x dx e 



