ON THE ATTRACTION OF A TERMINATED STRAIGHT LINE. [424 
of revolution having the extremities of the line for its foci, and that, if 
a shell bounded by any such surface and the consecutive similar surface, with 
equal to that of the line, then such shell and the line will exert the same 
attractions upon any point P exterior to the shell. The attractions of the line are 
obtained most easily by means of its potential ; viz. taking S, H for the extremities 
of the line, and, as above, the origin at the middle point, and the axis of x in the 
direction of the line, and writing 2ae for the length of the line, x, y, z for the 
coordinates of P, and r, s for the values of HP, SP (that is, r = V (x — ae) 2 + y 2 + z 2 , 
s= \/{x + ae) 2 + y 2 + z 2 ), then the potential is at once found to be 
32 
spheroid 
we have 
its mass 
V = log 
x + ae + s' 
x— ae+r ’ 
and we can hereby verify that the equipotential surface is in fact a spheroid of 
revolution having the foci S, H; for, taking the equation of such a spheroid to be 
X 2 y 2 + z 2 
a? + a 2 (1 - er) 
= 1, 
(a is an arbitrary parameter, since only the value of ae has been defined), we have 
and thence 
s = a + ex, r — a — ex 
x + ae + s = (1 + e) (x + a), 
x — ae + r = (1 — e) (x + a), 
and the quotient is =——- , a constant value, as it should be. The equation V= const, 
may in fact be written 
1 +e_x + ae + s 
1 — e x — ae+r’ 
viz. this equation, apparently of the fourth order, breaks up into the twofold plane 
X 2 y 2 + z 2 
y 2 = 0, and the spheroid - + ^ = 1. 
The foregoing results in regard to the attraction of a line are not new. See 
Green’s Essay on Electricity, 1828, and Collected Works, Cambridge, 1871, p. 68; also