82 
Proceedings of the Pioyal Society 
The potential due to any number of sources P i; P,„ and sinks 
P/ P 2 ', Ac., all of equal power, is got by simple superposition. If 
E be equal for all points, 
~ log r + jl log r , 
~ 
where r corresponds to a source, and r to a sink. Hence the equi- 
potential lines are 
r i r 2 r 3- 
= c 
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 dd 1 d0. 2 , Ac., at the sources, and d0[ d0 . 2 , Ac. at 
sinks, no fluid must flow. But the quantity of fluid per second 
reaching ds from P n is ^ E. The quantity withdrawn by P'„ 
Lit 
d6' 
is -—- E. Hence the differential equation of the stream-line is 
2 — 
- 'SdO' = 0 . 
Integrating, 2 9 - 2 9' = const. 
where 6 and O' are the angles between radii vectores and any fixed 
lines. If we agree to reckon 6 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 = ^ - PQP' = «. 
Hence the locus of Q is a circle through P and P', which is Kircli- 
hoff’s case. The orthogonals are circles whose centres (E) lie in 
PP' produced, and whose radii = \ PE.PH. 
(6.) If we have two equal sources and no sinks, or what is the 
