1904 - 5 .] Lord Kelvin on Deep Sea Ship-Waves. 1079 
ther of 
\wl = [ x(1 ~ tl)+2yt ’ 
And substituting t m , (either of these,) for t in (121) we find 
/ d 2 u\ 
Jl+t*. . .(124), 
or with simplification by (119), 
r ($) m = 2 ™™-^( 1 +C> 3/i (124) '- 
Eliminating t 2 m from the first factor of (124) by (122) we find 
which, with m — 1, and m — 2, gives /3 1 and /3 2 by (116). 
§86. Using (123) we see that (d 2 u/dij/ 2 ) m vanishes when x = y J8, 
and that it is negative for and positive for t 2 , when x>y */8. 
Hence t Y makes d 2 ujd\\r 2 negative. Therefore u 1 is the maximum ; 
and t 2 makes it positive. Therefore u 2 is the minimum ; and (119) 
gives for these maximum and minimum values 
ru 1 = (x + yt 1 ) Jl +t 1 2 , ru 2 = (x + yt 2 ) Jl+t 2 . (125). 
By (122), (123) we see that when yjx= 0, we have -t 1 = + oo , 
and — t 2 = 0. If we increase y from 0 to 4- x / J8, — 1 1 falls 
continuously from oo to and - 1 2 rises continuously from 
0 to J\. Thus - t x and - 1 2 become, each of them, ; which 
is the tangent of 35° 16'. 
§ 87. Geometrical digression on a system of autotomic , monopara- 
metric co-ordinates * §§ 87-90. 
In (119) put 
ru = a (126), 
where a denotes the parameter 0 W of the curve 0 C C, fig. 32, 
which we are about to describe ; being the curve given intrinsi- 
cally by (119) and (122) with suffix £ m 3 omitted from t. In the 
present paper these curves may be called isophasals, because the 
argument of the sine in (130) below is the same for all points on 
any one of them. 
Solving (119) and (122) for x and y, we find 
1 + 2^ / _ -t 
(l + * 2 ) 3/2 ’ y ~ a ( T+Wp 
■ (127). 
* Of this kind of co-ordinates in a plane, we have a well-known case in the 
elliptic co-ordinates consisting of confocal ellipses and hyperbolas. 
