324 Proceedings of Royal Society of Edinburgh. [ sess . 
§ 28. For fig. 10, instead of assuming as in (47) the calculation 
of Q(jb, t) for negative values of x, a very troublesome affair, we 
may now evaluate it thus. We have by (46) 
Q(u t) = t) - <f>(x +1,0 + </>0 + 2, 0 - 
Q( - X? t) = - X, t) - <f>( - X + l,t) + <£( - X + 2, t) 1 
Hence 
Q(x, t) + Q( - aj,. t) = <£(x, 0 - cf>(x + 1, t) + <f>(x + 2, t) - ) 
-<b(~x+l,t) + cf i (-x + 2,t)- P 
How by the property of </>, used in the first term of (54), that its 
value is the same for positive and negative values of x, we have 
<j>( - x + i, t) = cf>(x - i, t). Hence (54) may he written 
Qp 0 + Q( - x, t) = 2, ( - iy<K x + h 0 = 0 • (5-3). 
i= — ~j o 
Hence Q( - .r, t) = P(a?, 0 - Q($, jf) (56). 
Using this in (47) we find 
iP(M)-Q(M) ( 57 X 
for the elevation of the water due to the leftward procession 
alone at any point at distance x from 0 on the left side, x 
being any positive number, integral or fractional. Having pre- 
viously calculated Q(.c, t) for positive integral values of x , we 
have found by (57) the calculated points of fig. 10 for the leftward 
procession. 
§ 29. The principles and working plans described in §§ 11-28 
above, afford a ready means for understanding and working out in 
detail the motion, from t = 0 to oo , of a given finite procession 
of waves, started with such displacement of the surface, and such 
motion of the water below the surface, as to produce, at t— 0, a 
procession of a thousand or more waves advancing into still water 
in front, and leaving still water in the rear. To show the desired 
result graphically, extend fig. 10 leftwards to as many wave-lengths 
as you please beyond the point, I, described in § 24. Invert the 
diagram thus drawn relatively to right and left, and fit it on to the 
diagram, fig. 9, extended rightwards so far as to show no perceptible 
motion ; say to x = 200, or 300, of our scale. The diagram thus 
compounded represents the water surface at time 25 t after a com- 
mencement correspondingly compounded from fig. 8, and another 
