[sess. 1905-6.] Lord Kelvin on an Initiational Form. 
405 
The equivalence of (139) and (140) is easily proved by remarking 
that by (133) and (135), 
dF _ dF 
dx dz 
g dt 2 
(141), 
and therefore 
d 
{ED} 
1 
dx J(z + lx) 
~ 9t 2 l (?. 
£ 4(^)={ES} 1 a 
-1 
-gt 2 
g dt 2 J(z + lx) 
e 4 (z+LX) (142). 
§ 102. Look now to fig. 33, and see within how narrow a ( space, 
say from x — — 2 to x + 2 , in the new curve, the main initial 
disturbance is confined, while in the old curve it spreads so far 
and wide that at x — ± 20 it amounts to about *16 of the maximum 
disturbance in the middle, and according to the law of inverse 
proportion to square root of distance, which holds for large values 
of x for the old curve, at ^ = 80 it would still be as much as '1 of 
the maximum. The comparative narrowness of the initial dis- 
turbance represented by the new curve, and the ultimate law of 
decrease according to x"'i (instead of x~i for the old curve) are 
great advantages of the new curve in the applications and illus- 
trations of the theory to be given in §§ 135-157 below. 
§ 103. Kemark also that the total area of the old curve from 
— cc to + cc is infinitely great, while it is zero for the new curve. 
Eemark also that the potential energy of the initial disturbance, 
being 
igjdx[£(x, 1,0)P .. . . . (143), 
is infinitely great for the old curve, while for the new it is finite. 
§ 104. Equation (139) may be written in the following modified 
form, which is more convenient for some of our interpretations 
and graphic constructions : 
cos A . (144),. 
where - | X - tan" . (145). 
§ 105. The main curves, which for brevity we shall call water- 
curves in the accompanying six diagrams of fig. 34, represent the 
