Free Procession of Waves in Deep Water. 467 
§ 28. For fig. 10, instead of assuming as in (47) the caleu- 
lation of (Q(z, t) for negative values of x, a very troublesome 
affair, we may now evaluate it thus. We have by (46) 
Q(z, t)=1¢(2, ) —d(74+1, 4) + o(7+2,t)-—.... 
Q(—2, t)=49¢(—2, t)—d(—2+1, t)+6(—#+2,t)-.... 
Hence 
Q (a, t) + Q(—2, t)=¢(2,t) —O(@+1,1) + $(74 2, t)—... | (5A). 
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Now 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 $(—«# +7, t)=(v—i, t). Hence (54) may be 
written 
Qe, )+Q(—2,)= F (-1d(e+i,)=P(w, 0) . (58). 
Hence 
Ol ae aoe G2 ty. cys (OG) 
Using this in (47) we find 
PERO CHL): vnathee is fe 2) iy COLTS 
for the elevation of the water due to the leftward procession 
alone at any point at distance w from O on the left side, x 
being any positive number, integral or fractional. Having 
previously calculated Q(«, ¢) for positive integral values of z, 
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 ¢=0 to t=~, of a 
given finite procession of waves, started with such displace- 
ment 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 gra- 
phically, 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 e=200, or 300, of our 
seale. The diagram thus compounded represents the water 
2K 2 
