y 
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Waves produced by any given Initiating Disturbance. 6138 
rest. The displacements at any subsequent time ¢ are ex- 
pressed in real symbols by (7) (10) without the divisor ,/2, 
and by (8) (11) with a factor /2 introduced ; either of ae 
may be chosen according to convenience in calculation. One 
set has thus been calculated from (8), with g=4, and z=1, 
for six values of ¢; ‘5, 1°5, 2, 2°5, and 5; and for a suffi- 
ciently large number of values of 2 to represent the results 
by the curves shown in figs. 2 and 3. Except for the time 
t=5, each curve shows sufficiently all the most interesting 
characteristics of the figure of the water at the corresponding 
time. The enrve for £=5 does not perceptibly leave the zero 
line at distances x<1°8: but if we could see it, it would 
show us two and a half wavelets possessing very interesting 
characteristics ; shown in the table of numbers, $7 below, 
by which we see that several different curves with scales of 
ordinates magnified from one to one thousand, and to one 
million, and to ten thousand million, would be needed to 
exhibit them graphically. 
§ 6. Looking to the curves for ¢=0 and t=4; we see that 
at first the water rises at all distances from the middle of the 
_ disturbance greater than w=1°9, and falls at less distances. 
And we see that the middle (2=0) remains a crest (or 
positive maximum) till a very short time before t=}, when 
it begins to be a hollow. A crest then comes into existence 
beside it and begins to travel outwards. On the third curve, 
t=1, we see this crest, travelled to a distance z=1°7, from 
the middle where it came into being; and on the four th, fifth, 
sixth, seventh curves (figs. 1, 2) we a it got to distances 
zo, £8, 675, 22, at the times 13, 2, gat 5. This crest 
travelling rightwards on our diagrains has its anterior slope 
i very gradual down io the Satine level at z=c0. Its 
posterior slope is much steeper ; and ends at the bottom of 
the hollow in the middle of the disturbance, at times from 
t=ttot=14. At some time, which must be very soon after 
fi, this hollow begins to travel rightwards from the 
middle, followed by a fresh crest shed off from the middle. 
At t=2, the hollow has got as far as z="9 ; at t=234, and 5, 
: respectiv ely, it has Peat w=1'75 and ee re ‘Looking 
in imagination to the extension of our curves leftwards from 
the middle of the diagram, we find an exact counterpart of 
what we have been examining on the right. Thus we see an 
initial elevation, symmetrical on the two sides of a convex 
crest, of height 1:41 above the undisturbed level, sinking in 
the middle and rising on the two flanks. The crest becomes 
jess and less convex till it gets down to height 11, when it 
becomes concave ; and two equal and similar w ave-crests are 
