MATHEMATICAL THE OH Y OP STREAM-LINES. 
287 
it is obvious that the energy of current in an elementary space of the volume dx dy dz is 
^(u 2 +v 2 -\ -vf)dxdydz, . (A) 
and that the energy of disturbance is 
^(u 2 + v 2 -+- w 2 — 2u -f- 1 ) dx dy dz ( B) 
To find the total energy of current, or of disturbance, as the case may be, in a given 
finite space, the one or the other of the two preceding expressions is to be integrated 
throughout that space. In order to solve questions of this kind, recourse must be had 
to the velocity -function (<p) well known in hydrodynamics, and already referred to in § 2, 
equations (1) to (7), and in § 3, equation (9), as representing by its values a series of 
surfaces which cut all the elementary streams at right angles- — and especially to a pro- 
perty of that kind of function which was first demonstrated by Green, in his Essay on 
Potential Functions, and which is expressed as follows : — Let <p be a function of x , y and 2 , 
which fulfils the condition 
d 2 ? df? , n . 
dx 2 ' dy 2 ' dz 2 ’ 
(j 
let dir be an elementary part of the bounding surface of an enclosed space, and let ^ 
denote differentiation relatively to the normal to that elementary part, dn being positive 
outwards; then (under certain limitations which do not affect the subject of the present 
paper)* we have 
J ’JRS+ %■ +£) * % * =# t *. • 
(C) 
the double integral extending to all parts of the bounding surface. 
Observing now that 
dx dy 
dz ’ 
let E c denote the energy of current, and E D the energy of disturbance, within a given 
space, corresponding to the undisturbed velocity 1 and density I ; then we have 
E r 
(47) 
E D=lJjf>s A* dydz (47a) 
It is next to be observed that, because the velocity-function <p expresses a series of sur- 
faces cutting all the stream-lines at right angles, the coefficient (denoting the compo- 
nent velocity normal to the elementary surface da) is nothing for all bounding surfaces 
and parts of bounding surfaces that coincide with stream-line surfaces, — and therefore 
that, in finding the integral E c which expresses the energy of current within a given 
* As to the limitations to which this proposition is subject, see a paper by Helmholtz, in Crelle’s Journal 
for 1858, “TTeber Integrate der hydrodynamischen Gleichungen, welche den "VVirbelbewegungen entsprechen 
also Thomson en Vortex-Motion, Trans. Roy. Soc. Edin. 1867-68, pp. 239 et seqq. 
