MATHEMATICAL THEOEY OE STEEAM-LINES. 
289 
The energy of disturbance per unit of velocity and density is expressed by 
E d =B c - ft-p 1 -2*,+2* 1 + - (51) 
§ 15. Total Energy of Disturbance . — In order to find the total energy of disturbance 
throughout the indefinitely extended mass of liquid, the most convenient method is to 
find the limit to which E D in equation (47 a) approximates when the integrals are taken 
throughout a circular cylinder for cylindric cycnoids, or a sphere for cycnoids of revolu- 
tion, and the radius of the cylinder or of the sphere is indefinitely increased. The coeffi- 
cient is =0 at every point of the surface of the disturbing solid; therefore no inte- 
gration has to be performed over that surface. The triple integral in the second term 
of the equation, viz. 
— pj(w— \)dx dy dz , 
may be simplified by considering that, because the integration extends throughout the 
unlimited mass of the liquid, we have, by equation (46), 
ffl(u—l)dx dy dz— 0, 
and consequently 
— \\fii—f)dx dy dz— — l\fd$ dy dz. 
Now this is obviously the half difference, with the sign reversed, between the volume 
of the indefinitely large cylinder or sphere, as the case may be, and the displacement 
or volume of the disturbing solid, denoted by D. Moreover, in the first term of the 
equation, we have dn—dr ; \d<j—rdb for a cylinder, or 2-zr sin Qdb for a sphere, b being 
the angle which r makes with the axis of x ; and the limits of integration are from 4=0 
to b—lir for a cylinder, and from 5 = 0 to 4 =tt for a sphere. Hence we have the fol- 
lowing expressions : — For indefinitely deep cylindrical solids, 
for solids of revolution, 
E„=i C (<p 2 «•’ sin m+ (53) 
In taking the values of and I?- corresponding to an indefinitely great value of r, it 
is to be observed that the distance 2 a, or a ! + between a given pair of foci, becomes 
indefinitely small compared with r, and that consequently, if F be a function of the 
distance from a focus, and AF the difference of its values for a pair of foci whose distance 
apart is 2a, we are to make 
AF sensibly= — 2a-^- 
