126 Proceedings of the Royal Irish Academy. 



Taking the former alternative, on writing t = iq, r = iq, (15) becomes 

 {4,p~ + q^+ cp) sinh 22a sinh 2(ia - 2qq cosh 2c[a cosh '2c/ a + 2qq' cos 4pa = 0. 



(16) 



This may be written in the form 



il ~ ^'y sinh^(2 + q) a- {q + q')' sinh^(5' - q) a + 4:]r sinh 22a sinli 2qa 



- 4:qq' sin- 2pa = 0, (17) 



from which it is evident that it cannot be satisfied by real values of q, q' ; for 

 if they be chosen positive, as can always be done, the first term exceeds the 

 second, and the third the fourth. 



Falling back, then, on the latter alternative, and writing in (15) / = iq 

 simply, it becomes 



(4p^ + q'"^ - r-) sinh 2q'a sin 2ra - 2qr cosh 2qa cos 2ra + 2q'r cos 4^X1 = 0. (18) 



To find a stationary disturbance of given wave-length, and the correspond- 

 ing value of fi, we have then, supposing I given, to solve the simultaneous 

 equations involved in (18), and the statement that the values of m which 

 satisfy (11) are p ± r, - ^^ ± q'i. 



Now, from the coefficients of the powers of m in (11) we have the 

 equations 



if + g.''){f - ^^) = I', 



22J{q"' + r') = Bplfx-\ (19) 



If we express q, r, in terms of p, I, we have 



q' = 2pyf + P + f + l\ (20) 



r^ = 2p^/|/- + r- - f - l\ (21) 



and also obtain 



. = -^. (22) 



^P V p' + r 



It may now be proved that the equation (18) has no solution for which 

 2ra is less than tt. Denoting the left-hand member of that equation by V, 

 we have 



^d V/da = (^'2 + r'')(q cosh 2/a sin 2m - r sinh 2q'a cos 2ra) 



+ 4// {/ cosh 2q'a sin 2ra+r sinh 2q'a cos ra) - 4-pq'r sin 4j9a, (23) 



i d' V/da' = (2'- + rj sinh 2q'a sin 2ra 



+ 4:p\(q''- r-) sinh 2q'a sin 2ra+2qr cosh 2q'a cos 2ra - 2q'r cos 42Ja), 



(24) 



