96 



Proceedings of the Royal Irish Academy. 



The solution of (2), as an equation determining V, is easily expressible by 

 means of integrals, and is so expressed by Lord Kelvin. He does not, 

 however, make any reference to the problem of determining ^j so as to 

 satisfy assigned boundary-conditions. 



The most natural boundary-conditions to take would, of course, be that at 

 each of the bounding-planes u, v, w should vanish ; conditions which, as far 

 as V is concerned, are equivalent to the vanishing of V and dV/dy. The 

 analysis would obviously be much simplified, however, if two of the four 

 conditions which V can satisfy should be the vanishing of S at each of the 

 planes ; and it will be chiefly this case that I shall consider. It is readily 

 seen that we should have this case if the boundary-conditions were that 

 V should vanish, and that the tangential forces on the bounding planes should 

 be the same in the disturbed as in the steady motion. 



Denoting the bOunding-planes by y = ± a, instead of i/ = 0, h, as in 

 Part I., Chap. I., the equation determining the value of p evidently takes 

 the form 



ll 



'2 (//3 / vV'+j^ 



+ ai 



I-l 



\ O { V 



1(5 



•2(//3/ vX'+2^ 

 _S\v\~l(5 



^ L3 ( A ^ii 



- ai 



ai 



3 1-1 



= 0. (8) 



As the form of this is unaltered by changing the sign of i, complex roots 

 occur in pairs in the usual fashion. 



Art. 14. TM^ Period EcjuaMon has an Infinite Numher of Roots. 



In view of the suggestion of Lord Eayleigh,*' referred to above, it seems 

 desirable to prove, in the first place; that this equation in ]) has an infinite 

 number of roots; it has, in fact, an infinite number wdi(jse real parts are 

 negative. This may be shown by the aid of the approximate expressions for 

 the / functions for large values of the parameter. If we suppose that 

 (i'X- + p)//j3 has its real part negative, large compared with its imaginary 

 part, and large compared with a, we may take the argument of 



I'X" + ]J 

 to be a small positive angle, and that of 



+ ai 



See Art. 5,. p. 85. 



