82 
Proceedings of the Royal Society 
The potential due to any number of sources P 1} P 2 , and sinks 
P/ P 2 ', &c., all of equal power, is got by simple superposition. If 
E be equal for all points, 
u = C - 2 A log r 4- 2 A log r' , 
where r corresponds to a source, and r' to a sink. Hence the equi- 
potential lines are 
= G . . . . (5). 
r i r . 2 r 3 ... 
The equation of the lines of flow follows at once from the equa- 
tion of continuity. Across any element ds of a stream line sub- 
tending angles d6 1 d0 . 2 , &c., at the sources, and d0 2 d0. 2 , &c. at 
sinks, no fluid must flow. But the quantity of fluid per second 
reaching ds from P » is E. The quantity withdrawn by P' n 
2i7T 
dO' 
is -—A E. Hence the differential equation of the stream-line is 
2 dd - 2 dO’ = 0 . 
Integrating, 2# - 20' = const. 
where 6 and O' are the angles between radii vectores and any fixed 
lines. If we agree to reckon 0 in opposite directions for sources 
and sinks, the equation becomes 
2# = a . . . . (6). 
The following are elementary consequences of this equation : — 
(a.) When we have one source P and one equal sink P', the 
stream line through any point Q has for its equation 
20 = QPP' + QP'P = X - PQP' - a. 
Hence the locus of Q is a circle through P and P', which is Kirch- 
hoff’s case. The orthogonals are circles whose centres (R) lie in 
PP' produced, and whose radii = VPR.PTt. 
( b .) If we have two equal sources and no sinks, or what is the 
