Free Procession of Waves in Deep Water. 455 
have, for the initial height of the free surface above the 
undisturbed level, 
Z bs Zz 
—{=¢(2, naa/ i feet? - (18). 
§ 13. We shall first take, as initiating disturbance, a row 
extending from —x to + of superpositions of (18) : 
alternately positive and negative; and placed at equal 
successive distances 4X: so that we now have 
—h=Pia, N= = (-4(e+i, 0). oh (hoy 
or, as we may write it, ac 
—=P(x, 0)= Ss Dieses Oaiiao 193 
where beg 
D(x, 0)=(@, 0) -—d( e+ a 0) Dr OE 
In (19), P denotes a space-periodic function, with » for its 
period. This formula, with ¢ substituted for 0, represents 
—, being the elevation of the surface above undisturbed 
level at time ¢, in virtue of initial disturbance repre- 
sented by (19). 
§ 14. Remark now that whatever function be represented 
by @, the formula for P in (19) implies that 
P Eee Oz bia, Oy 2k 1 QE), 
which means that P is a space-periodic function with » for 
period. And (19) also implies that 
Rie eo = P(#, 0). . %~ = (22); 
which includes (21). And with the actual function, 
(18), which we have chosen for (x, 0), the fact that 
d(x, 0) =¢(—2, 0) makes 
Bei E(— 47, Oe. yn x...< (23). 
Thus (19) has a graph of the character fig. 5, symmetrical 
Fig. 5. 
on caeh side of each maximum and minimum ordinate. The 
