1904 - 5 .] Lord Kelvin on Deep Sea Ship- Waves. 
1081 
§ 90. Going back now to § 86 and the continuous variations con- 
sidered in it, we see that - 1 1 and - t 2 are respectively the tan- 
gents of the inclinations, reckoned from OY clockwise, of portions 
of the long arc 0 C and of the short arc W C, in the upper half of 
the diagram. Thus, if we carry a point from 0 to C in the long 
arc, and from C to W in the short arc, we have the change of in- 
clinations to OY represented continuously by the decrease of 
tan -1 ( - L) from 90° to 35° 16', while y increases from 0 to x JS ; 
and the farther decrease of tan -1 ( - t 2 ) from 35° 16' to 0°, while y 
diminishes from xJ8 to 0 again. The inclination to O Y of the 
two branches meeting in the cusp, C, is 35° 16' (or tan -1 N /J). 
For any point in the short arc GWC of the curve u or 
cos (if/ - 0)/cos 2 if/, is a minimum. In each of the long arcs u is a 
maximum. At every point of the curve the value of u, whether 
minimum or maximum, is a/r. Hence for different points of the 
curve, u is inversely proportional to the radius vector from O. 
§ 91. Going back to (118)' we now see that for all points on 
any one of our curves, ru x and ru 2 have both the same value, being 
the parameter O W of the curve. The first part of (118)' is one 
constituent of the depression at any point on either of the long 
arcs; and the second part of (118)' is one constituent of the 
depression at any point on the short arc. Taking for example 
the largest of the curves shown in fig. 32, we now see that for any 
point of either of its long arcs, the second constituent of the 
depression of the water is to be calculated from the second part 
of (118)'; while for any point of its short arc, the second con- 
stituent of the depression is to be calculated from the first part of 
(118)'. 
§ 92. Explaining quite similarly the determination of d(&, y) 
for every point of each of the smaller curves which we see in the 
diagram cutting the longer arcs of the largest curve, we arrive 
at the following conclusions as the complete solution of our problem. 
The whole system of standing waves in the wake of the travel- 
ling forcive is given by the superposition of constituents calcu- 
lated according to (127). with greater and smaller values of the 
parameter a with infinitely small successive differences. Hence, 
what we see in looking at the waves from above is exactly a 
system of crossing hills and valleys, with ridges and beds of 
