620 Deep-Water Two-dimensional Waves. 
§ 10. Our assumption h=1 for the free surface involves 
no restriction of our solution to a particular case of the 
general formula (7). Our assumption g=4 merely means 
that our unit of abscissas is half the space fallen through 
in our unit of time. The fundamental formulas of § 3 may 
be geometrically explained by, as in $2, taking O, our 
origin of co-ordinates, at a height h above the water level, 
and defining p as the distance of any particle of the fluid 
from it. When, as in §§ 5-9, we are only concerned with 
particles in the free surface (that is to say when z=h), we 
see that if 2 is a large multipie of z, p==a. See for example 
the heading of the table of § 9. And if we are concerned 
with particles below the surface, we still have p==a, if x is a 
large multiple of z. Thus we have the following approxi- 
mation for (7) of §3:— 
2g — gt? z 
: ed! gt? _. g? | = 
iP(zZ, 2, = ya | v (e+e) cost +V7 (e—z) sin | « (lal 
Suppose now d@/di to represent ¢ (instead of d, asin §§ 5-9) ; 
we have 
d 
which is easily found from (13) without farther restrictive 
suppositions. But if we suppose that z is negligibly small 
in comparison with 2; and farther that 
<==) 1. 4) 
we find by (14) 
witha a ( GE: the #) 
—-—-a5(Cos_——sin=—] , Jy . 
eo Zin 2.42? Ae y Av ee 
This, except the sign — instead of +, is Cauchy’s solution *; 
ot which he says that when the time has advanced so much 
as to violate a condition equivalent to (15), ‘‘ le mouvement 
change avec la méthode d’approximation.” The remainder 
of his Note XVI. (about 100 pages) is chiefly devoted to 
very elaborate efforts to obtain definite results for the larger 
values of t. This object is thoroughly attained by the expo- 
nential factor in (8) of §3 above, without the crippling 
restriction z/v==0 which vitiates (16) for small values of z. 
* Guvres, vol. i. note xvi. p. 193. 
