208 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Omitting for the moment the equation M = 0, we have the four 
equations 
4. V4.1v 1 
/3 T ^ T 
r fss 
= vf ’ 
/3 r a 
while 
y = r sin (nx ,2 + k ') . 
From the second and the third of these, we have 
and therefore 
so that 
consequently 
(/3rn'-rfti') = |'(rft'-/3r,'), 
(r/3f - /3T-^')S = constant = 2k , 
& _ o/A . 
(B V /3V8 ^ a ^ 
where L is Si constant. Thus 
/3 = (Zm)^a^a/“^ , r = . 
When these values of /3 and V are substituted in the first of the foregoing 
equations, it becomes 
1 V ^ _ T.2 A^ _ _ a V • 
'^a/2 a2 a 
that is, an equation involving S alone, the independent variable x^' being 
given by the definition 
adx^ = dx-^. 
Thus far, the second and the third equations have been used only in a 
single combination ; so we must substitute the values of /3 and r in either 
of them. Now the value of /3 gives 
and therefore 
By the third equation. 
/3 s r VS/ ^ 8 ’ 
