1905-6.] Lord Kelvin on the Growth of a Train of Waves. 431 
distribution of disturbance through any finite space however great 
on each side of 0, left to itself without any application of surface- 
pressure, becomes dissipated away to infinity on the two sides ; 
and leaves, as illustrated in §§ 96-113, an ever-broadening space 
on each side of 0, through which the motion becomes smaller and 
smaller as time advances. 
§ 150. It remains only to look into some of the analytical 
details concerned in the practical working out of our solutions 
(189), (190). Taking cos a >(t — q) in the formulas, and taking case 
<h, we find by (190) 
Sfx , z , t) = P cos wt + Q sin wt , , . (196): 
where P = dq cos ioqcj>(x ,z,q); and Q= 1 dq sin wqc^{x , z , q) (197). 
Jq -A) 
When P and Q have been thus found by quadratures, for all 
values of t, and any particular value of x, by integration by parts 
on the plan of § 134, we readily find, without farther quadratures, 
or integrations, expressions for the seven other formulas included 
in (189), (190). 
§ 151. Let us first find P and Q for t — co. Using the 
exponential form for <£ , given by (137), we find 
P={RS}^/— J dq cos coqe~ mg2 ; and Q = {RS}^/— j dq sin o)qe~ mq2 (198) 
where m = -^g/(z + ix). 
Hence, according to an evaluation given by Laplace in 1810,* w T e 
find, taking g = 4, 
-W2 
P={ES} A /|e 4 ” ( 199 ). 
The definite integral for Q is a transcendent function of w and m, 
not expressible finitely in terms of trigonometrical functions or 
exponentials. By using the series for sin c oq in terms of (a >#) 2i+1 , 
and evaluating I dq q 2i+1 e~ q2 by integrations by parts, we find the 
* o 
following convergent series for the evaluation of Q, for t = ca ; 
and g = 4 : — 
Memoires de PInstitut, 1810. See Gregory’s Examples, p. 480, 
