196 
Proceedings of Royal Society of Edinburgh. [sess. 
§ 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 he geometrically explained 
by, as in § 2, taking 0, our origin of co-ordinates, at a height k 
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 a? is a large multiple of z, p=^x. See for 
example the heading of the table of § 9. And if we are concerned 
with particles below the surface, we still have p=.x, if a is a 
large multiple of z. Thus we have the following approximation 
for (7) of § 3 
z, t) ' — — 
Zi.X 
Suppose now dcfr/dt to represent 
we have 
J(x + z) cos 
4x 
+ J(x - z) sin 9L - J € 4 * 2 (13). 
£ (instead of <f>, as in §§ 5-9) ; 
i=J t !<#>(*, 2,0 
• (14), 
which is easily found from (13) without farther restrictive 
suppositions. But if we suppose that z is negligibly small in com- 
parison with x ; and farther that 
6^-0 
4a; 2 
we find by (14) 
. (15), 
gt 
'2J2.xW 
at 2 . gfi 
cos — — sm — 
. 4% 4x 
■ (16). 
This, except the sign - instead of +, is Cauchy’s solution;* of 
which he says that when the time has advanced so much as to 
violate a condition equivalent to (15), “le mouvement change 
“ avec la methode 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 exponential factor in (8) of § 3 
above, without the crippling restriction zjx = 0 which vitiates (16) 
for small values of x. 
* (Euvres, vol. i. note xvi. p. 193. 
(. Issued separately April 4, 1904.) 
