Free Procession of Waves in Deep Water. 461 
Fig. 10 represents the free surface thus found for the 
leftward procession alone at ne C= 20m: 
§ 25. The function D(z, t), which appears in § 13 as an 
en in one of the modes of summing shown for P(a, 0) 
in (19’), and indicated for P(«, ¢) at the end of § 13, and 
which has been used in some of our summations for Q(2, ¢); 
is represented in figs. 6 and 7, for t=0,and t= 247 respectively. 
§ 26. Except for a few of the points of fig. 6, representing 
D(w«, 0), the caleulation has been performed solely for integral 
values of 2. It seemed at first scarcely to be expected that a 
fair graphic representation could be drawn from so few calcu- 
lated points; but the curves have actually been drawn by 
Mr. Witherington with no other knowledge than these points, 
except intormation as to all zeros (curve cutting the line of 
abscissas), through the whole range of each curve. The 
calculated points are marked on each curve; and it seems 
certain that, with the knowledge of the zeros, the true 
curve must lie very close in each case to that drawn by 
Mr. Witherington. 
§ 27. The calculation of Q(a, ¢), for positive integral values 
of wv, is greatly eased by the following arrangements for 
avoiding the necessity for direct summation of a sluggishly 
convergent infinite series shown in (46), by use of our 
knowledge of (Pz, ¢). We have, by (46) and (19), 
Q(0, t) =. 6(0, t)— (1, t) +4(2, t)—....ad inf. (48), 
P(O, N= = (-1)94, Rat eae 
Hence, in virtue of ¢(—z2, t)=(2, ¢). 
eye, 6) 4 ek 2. (50), 
Again going back to (46), we have 
()(z, t) =$6(2, t)-—d(et+1, t)+6(4+2, t)-—d(e+3, t)+.... 
Q(e+1,t)= 3o(e+1,t)—(e+2,t) +o(e4+3, t) — 
By adding these we find 
Q(a +1, é) + Q(z, t)=4][ d(2, ) —d(#+4+1,#) ]=4D(«, t).(51). 
By successive applications of this equation, we find 
2Q (x +4, t) =(—1)2Q(2, t)—(—1)'D(a2,t)+...4+ Diet+i—1,t) (52). 
Hence by putting «=0, and using (50), we find finally 
2Q (2, 4) = (—1)*P(0, 4) —(—1)*D(0, 4) +...4+ Di—1, t) (53). 
This is thoroughly convenient to calculate Q(1, ¢), QQ, ¢)... 
successively; for plotting the points shown in fig. 9 
