MATHEMATICAL THEOEY OF STREAM-LINES. 
295 
For the disturbance caused by a sphere, half immersed, equation (67) takes the fol- 
lowing form, 
(71) 
Z =r 
2 cos 2 9 — 1 
• 3 cos 2 6 — 1 ’ 
in which r is the horizontal distance from the centre of the sphere, and 0 the angle that 
r makes with the axis of x ; and the same value of Z is approximated to at distances 
from a disturbing solid of any figure which are very great compared with the dimensions 
of the solid. 
The following examples are calculated for the oval neoicl of revolution LB, and for 
the cycnoid of revolution L'B', shown in fig. 3, the unit of measure being one tenth part 
of the distance from the axis OX to the nearest of the straight lines that are parallel to 
it, — also for a sphere of the radius 1. 
Oval. 
Cycnoid. 
Sphere. 
Half-length l ....... 
64 
95 
1 
Extreme half-breadth y a = greatest! 
depth of immersion ./ 
26 
31-6 
1 
Mean virtual depth Z m ... . 
19-1 
20-5 
2 
3 
Virtual depth at ends Z, . 
13-2 
13-5 
1 
2 
Virtual depth amidships Z yo . 
55-5 
62-4 
1 
When a wave of a given length travels in water of unlimited depth, the virtual depth 
of disturbance is equal to the radius of a circle whose circumference is equal to the 
length of the wave. For a wave of a given ‘periodic time , in water of unlimited depth, 
the virtual depth is equal to the height of a revolving pendulum which makes one revo- 
lution in the period of a wave. For a wave travelling at a given speed , under all cir- 
cumstances whatsoever, the virtual depth is twice the height due to the speed ; and con- 
versely, for a given virtual depth , under all circumstances, the speed is that acquired 
during a fall through half the depth. (See Proceedings of the Royal Society, 16tli 
April, 1868, page 345.) These laws are expressed as follows. Let W be the speed of 
advance of a wave in a horizontal direction perpendicular to the line of its crest, X its 
length, T its period ; then we have 
in water of any depth, limited or unlimited, 
W=gZ; (72) 
and in water of unlimited depth, 
T=2^v^; (72 a) 
a = W T = 2 tZ ; . . (72 b) 
and therefore 
(72 c) 
w=«T. 
2 7T 
(72 d) 
