Growth of a Train of Waves. 31 



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 o£ our solutions 

 (189), (190). Taking cos (o(t — q) in the formulas, and taking- 

 case 0, we find by (190) 



S^a?, z, i) =¥ cos a,t +Q sin cot . . (196); 



where P=i dq cos <oq<f>(x, z, q) ', | 



J ° , t \ (197). 



. and Q=l dq sin G}q<f>(x, 2, q) 



Jo 



When P and Q have been thus found by quadratures, for all 

 values of t, and any particular value of #, 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= go. Using the 

 exponential form for <£, given by (137). we find 



P={B,SU /— P^coso^e-™? 2 ; ] 



' /Jo /smr r (198) > 



and Q={RS}^/— J dq sin <*>qe-™<? j 



where 7n= Tyl( z + £ti ') • 



Hence, according to an evaluation given by Laplace in 1810 *, 

 we find, taking tf=4, 



P={RS}/y/|e^ .... (199). 



The definite integral for Q is a transcendent function of eo 

 and in, not expressible finitely in terms of trigonometrical 

 functions or exponentials. By using the series for sin coq in 



terms of (o)g) 2l+1 , and evaluating 1 dq q 2i+1 e~ q2 hy integra- 



Jo 

 tions by parts, we find the following convergent series for 

 the evaluation of B, for t=co ; and g = 4z : — 



* Memoires de V Institute 1810. See Gregory's 'Examples/ p. 480. 



