97 
of Edinburgh, Session 1869 - 70 . 
cal surfaces, we must take as starting point, not a single source, 
but a source and sink at the extremities of a diameter. For 
brevity, we shall speak only of the source, assuming the existence 
of a corresponding sink. 
When there is one source, the stream lines are manifestly great 
circles through it, and the equipotential lines small circles, of which 
it is the pole. 
If the radius of the sphere is a , the circumference of the small 
circle, whose angular radius is 0, is & ra sin 6. Hence if u be the 
potential, 
du 
dd 
u 
For any number of sources the potential will be 
-fs ± log l~ cos e j, 
2 \ ^l-s-cosfl/’ 
and the equation of the equipotential lines, 
1 - cos 0 X 1 - cos _ q 1 - cos 0\ 1 - cos 
1 + cos 9 l ’ 1 + cos 0% 1 + cos 9\ * 1 + cos Q'f" ; 
the accented angles belonging to sinks. 
For the lines of flow we have, precisely as in a plane, 2(=fc (p) = c, 
where <p is the angle between the great circle through a source 
and a point on the line, and a fixed great circle through the source. 
Let us take, as an example, the case of one source and one sink. 
Let the co-ordinates of these points be h, k, 0 ; h , - h } 0, and those 
of any point on an equipotential line, x, y , z. 
We have for the equation of this line, 
1 - cos 0 + ^ 1 - cos $' 
1 + cos 6 1 + cos & 
where 
hx + ky 
cos 6 = , cos 6 = 
a 2 
Hence the projections of the equipotential lines on the plane of 
xy have as equation, 
(a 1 -hx- Tcy) ( a 2 + hx — hy) + X (a 3 - hx + ky) (a 2 + hx + ky) = 0, 
= o, 
hx — ky 
1 , 1 - cos 9 
2 log r^T0 
