MATHEMATICAL THEORY OF STREAM-LINES. 
291 
for solids of revolution, 
^2=l-^.2 k'a-, (60 a) 
observing, in the last expression, that for any pair of foci whose distances from the origin 
a ' and a" are unequal, the mean of those distances, — - } — , is to be taken as the value of a. 
When the disturbing solid is a sphere of the radius Z, its displacement is 
It has one focus at its centre, produced by the coalescence of a pair of foci ; P becomes 
indefinitely great, and a indefinitely small ; but their product has a finite value, 
Z 3 
h?a=—. Hence in this case we have 
2 
9TC 
^=i-W; (60 b) 
that is to say, the total energy of the disturbance produced by a sphere is equal to half 
the energy of the sphere. 
When the solid is an oval or bifocal neo’id of revolution, and the excentricity a increases 
indefinitely as compared with the parameter P, the displacement approximates upwards 
towards that of a cylinder of revolution of the length 2 a and transverse section 27 rk 2 
(that is, towards 47 rlc 2 a) ; so that in this case we have for the upper limit of the ratio of 
the total energy of disturbance to the energy of the solid, the following value : — 
^ = W=f (60 c) 
For all neoicls of revolution, oval and cycno'icl, the ratio in question lies between the 
limits \ and § . Its value in any particular case may always be determined to any 
required degree of approximation by constructing the figure of the disturbing solid and 
measuring its displacement. For example, in fig. 3 it is found to be, for the oval neoicl 
of revolution LB, 0 - 56 ; and for the cycnoid of revolution L' B', 06 nearly. 
The principles of this and the three preceding sections (§ 12, 13, and 14) are appli- 
cable not only to bifocal, quadrifocal, and other stream-line surfaces having foci situated 
in one axis, but to all stream-line surfaces which can be generated by combining a uni- 
form current with disturbances generated by pairs of foci arranged in any manner what- 
soever, or having, instead of detached focal points, focal spaces ; the disturbance-functions 
belonging to which are to be found by integrating the corresponding functions belonging 
to the points contained in those spaces, a process similar to that of finding the potential 
of a solid*. 
§ 16. Disturbance of Pressure and Level . — It is well known that in all cases of the 
steady flow of a liquid, the sum of the height due to velocity, and the height due to 
elevation and pressure combined, is constant in a given elementary stream ; that is to 
* Note by the Reporter. — See paper, Professor C. Neumann, in Ckelle’s Journal for 1861, on the equation 
d 2 u d 2 u 
dx 1 dy 2 
2 s 
MDCCCLXXI. 
