294 
PEOFESSOE W. J. MACQUOEN EANKINE ON THE 
when the column is a prism of finite dimensions, the mean virtual depth is as follows : — 
z _ fff ^-\)dxdydz ( 66 ) 
m — 1 )dxdy 
When the disturbance is that produced by the longitudinal advance of a solid whose 
figure is a stream-line surface of revolution, with any number of pairs of foci, floating 
immersed to the axis in a liquid of indefinite depth, the integrations indicated in the 
preceding equations give the following results : — 
Virtual depth at a given point, 
**{*?+¥) 
2e(^+2f) : 
(67) 
the notation being the same as in equations (48) and (49). 
Mean virtual depth throughout the whole mass, 
( 68 ) 
in which D is the displacement of the floating solid, and S the area of its water-section 
(that is, of its horizontal section in the plane of the surface of the water) ; so that the 
mean virtual depth is equal simply to the mean depth of immersion of the solid. Here 
it must be explained that, when the disturbances relatively to the centre of mass of the 
liquid are integrated, equation (68) takes the form Z m =%, and that the value ^ is 
obtained by taking the disturbances relatively to the common centre of mass of the 
liquid and solid. 
At the two ends of the floating solid, where x=l and y=. 0, the virtual depth takes 
the following value, 
Z|= — 
2/S 
k q a 
k^a 
•(/ 2 -« 2 ) 2 
(69) 
When there is but one pair of foci, this is reduced to 
At the midship section (B and B' in the figures) the virtual depth is 
k*a 
Z 
_ «*+y 2 
Jra 
Vo 
(70) 
(« 2 + Z/o)« 
y 0 being the extreme half-breadth. When there is but one pair of foci, this becomes 
simply % / a 2 +y 2 0 . 
