196 The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



Hence if /, , ?», , n, , h , m, , 7i2 , k,m^,ni be the three systems of values of I, /n, n, 

 we shall have 



rii cos ^1 cos wi, + <I>i cos^ m, + ^i cos wi, cos n, 



— Ila COS^ ^i + <I>2 COS U cos 7Hi + •^2 COS /[ COS ?Z, , 



n, cos 4 cos ?W2 + 4>, COS- mj + •*^i cos m^ cos Wj 

 = rij cos- 4 + <l>2 COS ^2 cos rwj -^ ■*'2 cos ^2 cos Wj, 



fl, cos 4 cos H(3 + Oi COS- J/Jj + >I^, COS 7«3 COS U^ 



— rij COS- 4 + O2 COS 4 COS m-i + ■*■, cos 4 cos n-^. 



Adding these equations, and recollecting that, as the three directions of vibra- 

 tion are rectangular, 



cos^ 4 ■\- cos- 4 + cos^ ?3 = 1, 



cos^ '»!, + cos^ mo + cos^ m^ — 1, 

 cos Ix cos m, + cos L cos Wa + cos Z3 cos wis = 0, 



cos Ix cos Ki + cos 4 COS Ko + COS 4 COS Ms = 0, 



cos vix COS n, 4- cos to, cos n.^ + cos wjs cos 713 = ; 



we have 



and similarly 





or, substituting for n.,, &c., their values from (0'), 



(^,v - ^av) «' + (^l^v - ^/i' .0 ^' -1- (^7V - A v) cH 2(^^^, - 5^,„.) ?"^ 



+ 2(^^,, - (7,,,,) 6c + 2(5„„. - C„,,,) ac + 2(5„,,. - C^) = 0, 



+ 2(C,,„. - A,^,) be + 2( C„,.. - ^„„0 ac + 2( C„,„, - ^„,,,) at = 0. 



If these equations hold for all directions of wave plane, it is easily seen that 



the coefficients of 



a-, b-, c", ab, ac, be, 



must vanish of themselves. This condition will give eighteen equations which 



