288 
PEOFESSOE W. J. MACQUOEN EANK1NE ON THE 
space, it is necessary to take into account those boundaries only of that space which inter- 
sect the stream-lines. 
The values of the velocity-function <p for quadrifocal stream-line surfaces are obviously 
the following : — For cylindrical stream-line surfaces, 
hyp log hyp log^- 4 ; ■ • • (48) 
' 1 '3 
for stream-line surfaces of revolution, 
is. 
(49) 
in which r x , r 2 , r 3 , and r 4 denote, as before, the distances of a point from the four foci. 
These expressions may be made applicable to bifocal surfaces by making ^=0, and 
might be extended to surfaces with any number of pairs of foci by increasing the number 
of terms and parameters. 
When a pair of foci coalesce, the function of r belonging to those foci is to undergo 
the operation — A in which A is an arbitrary constant of one dimension — thus giving, 
kkx 
for cylindrical surfaces, a term of the form 2 -, and for surfaces of revolution a term 
of the form 
A*. Ax 
2?- 3 ‘ 
In the foregoing investigations, and in their appplications which are to follow, the 
energy of disturbance is taken relatively to the centre of mass of the liquid. If taken 
relatively to the common centre of mass of the liquid and solid, it would be increased by 
a quantity whose value for the whole mass of liquid, per unit of undisturbed velocity 
and of density, is 
2(L+D) 2 ~2(L + D) 2 t |jy < ^ ^ z » (49 b) 
but when the extent of the liquid is unlimited, that quantity vanishes as compared with 
the quantity given by equation (47 a). 
§ 14. Energy in an Elementary Stream. — In order to apply the principles of the pre- 
ceding article to the whole or to a given part of an elementary stream, let <r 0 be the 
transverse sectional area of that stream when undisturbed, measured on a plane normal 
to x, <7 the sectional area on such a plane at a given point, x x and x 2 the values of x, 
< 7 , and <r 2 the values of a, and <p, and (p 2 the values of cp, for the two ends of the part of 
the stream under consideration; and let x 2 be greater than x x . Then the energy of 
current, per unit of undisturbed velocity and of density, is found by taking the integral 
in equation (47) for those two ends only; that is to say, 
Eo = ifc 2 s 
1 dx x 1 ’ 
but ^<7=«<r=<7 0 ; and therefore we have simply, for the energy of current, 
E c=?(&-p,)- 
(50) 
