43 Prof. T. H. Havelock. 
We can now apply the surface condition (7) by substituting the expansions 
(10) and (13) and equating coefficients of cos ; COS 3¢, .... 
We obtain, as in Stokes’ method, an infinite series of equations of the 
form A2.+1 = gBon+1, from which the quantities g, Bi, Bs, B3, ..., are to be 
obtained in practice by successive approximation from the equations taken in 
order. 
Up to terms of the third order the equations are 
1+ 58 + 1282+ 1081? + 288; 82+ 58:2 +1883 
—6 (61+ 282+ B?+58:B2+Be+ 383) cosh 2a 
= 9 (Boi + 61Bu + GsBa, + BBs), 
5Bi+ 862+ 481? + 2881 8.+ 5813+ 1883—6 (282+ Bi?+ 381 B2+ 383) cosh 2a 
= g (Bos + A:Bi3+ 82Bo3 + B3B33), 
882+ 481? + 118182+1183;—18 (8182+ 3) cosh 2a 
= 9 (Bos + BiBis + B2Bo5+ B3Bs5), 
1183+ 118182 = 9 (Bor +AiBiz+ B2Boz + B3B37). (15) 
It might appear, from the quantity cosh 2a on the left, and from the 
factor e* in the expressions for the B’s in (14), that there are terms in these 
equations which become infinitely large as « increases indefinitely. But we 
have 2, = bye~**, Bs = doe 42, B3 = bse8*, ..., therefore, if we write the 
equations (15) as a set of equations for the coefficients 6), bs, ..., this 
difficulty disappears. In this connection we may recall the initial assump- 
tion that the series in (5), namely, 1+ b,c? + hyetiv 4 ig absolutely 
convergent, otherwise the analysis has no meaning. 
The infinite set of equations, given to the third order in (15), has to be 
treated by Stokes’ method; that is, assuming the process to be convergent, 
the equations taken in succession yield approximations to g, Bi, Bs, ..., for any 
assigned numerical value of «. But there is a difference between these 
equations and the corresponding set in Stokes’ analysis. In the latter, the 
first coefficient, say 6, is arbitrary, and the successive equations have their 
lowest terms of order zero, one, two, and so on, respectively ; thus g and the 
remaining coefficients are found as power series in }. But in (15), we have 
a term independent of the 4’s on the right-hand side of each equation ; thus 
the solution, if practicable, leads to a set of numerical values of g, Gi, Bo,... 
for a definite numerical value of «. We may notice, in passing, that for a first 
rough approximation gBo: = 1; and as By, does not differ much from unity 
for any value of a, the coefficients 81, Be, ..., are of the order of magnitude of 
Boz, Bos, ---; respectively. 
4. The method of approximation used in the following calculations may 
be described by considering first the simplest form of the equations, namely, 
137 
