IN CRYSTALLIZED MEDIA. 34S 



Now h/, = {Lv cos 62 + ^y cos (p^ + ox cos \//,,y ; 



••• 2 %^ sin^ ^ = itf cos' e, + N cos^ <^, + P cos^ ^, 



, „ 1 . 2 k^x 

 and 2 - sm — — = c ; 



when M= 2 -— sni' ——- , N= S ^ sni'' -— , 

 r" 2 r' 2 



P=2-— sin^— — ; 

 r" 2 



and .-. v' = 2 {(c - 3 M) cos^ e, + {c - 3 N) cos' (p, + (c- 3 P) cos' ^{^2} 

 = a" cos' 0o + b- cos- (^.3 + c^ cos' x//2 *, 



«-, b-, (? being respectively the squares of the velocity of transmission, 

 when the vibration is simply in the direction of one of the axes. 



Similarly v"' = a- cos^ ^3 + b' cos" ^3 + c' cos \f/3. 



17. Now we have seen that 



S -^ — sm — — - =0. 



r' 2 



Applying the same process to this equation, we obtain 



M cos 02 cos 03 + N cos (p2 cos (pt + P cos ^^ cos -^-^ = 0, 



which will determine the directions of vibration corresponding to tlie 

 values of the squares of the velocities given above. 



18. In order to determine these directions, we express the co-ordinates 

 and angles by making e the angle between the axis of x and the line of 

 intersection of the planes of yi», with xy; and calling m the angle between 

 this line and the axis of y, -^ the angle of inclination of si,y, to xy. 



Thus xPN= e, (see note at end of Sup.) 



NPy,=,j.. 



• This equation I have obtained by a totally different process in a subsequent Memoir. 



