Lord Kelvin on an Initiational For 



m. 



The equivalence of (130) and (140) is easily proved 

 remarking that by (133) and (135), 



and therefore 



~r= L i =~ n« .... (141), 

 die ctz a at- 



4 RD I d ~ 1 e^ lx) - J 1*8 ll - ~ ~~ X e 1 ^ (142). 



§ 102. Look now to fig. 33, and see within how narrow a 

 space, say from #= — 2 to .i'=+2, in the new curve, the 

 main initial disturbance is confined, while in the old curve it 

 spreads so far and wide that at ,r = +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 com- 

 parative narrowness of the initial disturbance represented by 

 the new curve, and the ultimate law of decrease according to 

 x'* (instead of x~* for the old curve) are great advantages 

 of the new curve in the applications and illustrations of the 

 theory to be given in §§ 135-157 below. 



§ 103. Remark also that the total area of the old curve 

 from —x to + x is infinitely great, while it is zero for the 

 new curve. Remark also that the potential energy of the 

 initial disturbance, being 



At[«*,l,0)]» .... (143), 



— DC 



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. 



-g&z 



TJr(z, z, t)= - A ~^n^ cos A v U4 )> 



where A= V^ — - y — tan -1 —f ^v (14o). 



1/r 2 ^ at- cos x~-P 



§ 105. The main curves, which for brevity we shall call 

 water-curves in the accompanying six diagrams of fig. 34, 

 represent the surface displacements according to our new 

 solution -v|r(a', :, t) for the six values of t respectively, 0, \ *Ji7 y 



