52 ROYAL SOCIETY OF CANADA 
The last occurs when ay’ si ay’ = 0 onacertaininterval. Thisare, if 
produced, passes through O. We also suppose that lim. exists! as we 
approach the attracted particle; this is equivalent to assuming the exist- 
ence of the tangent at this particle. We do not, however, assume the 
existence of a tangent at any other point. We assume also that the 
meridian curve has no double points. This last hypothesis, as remarked 
in Chap. I, $6 (b), enables to distinguish arithmetically the interior and 
exterior of the bounding surface. Without it we may indeed show that 
the attraction integral may be made intinite without violating the isoperi- 
metric condition.” 
(b). Inasmuch as the matter is homogeneous, the mass will be con- 
stant if the volume is. A convenient method of obtaining the latter, 
and also the attraction integral, is to find the effect of small conical shells, 
vertex the origin, cut out by right circular cones about the axis of revo- 
lution. If A denote thearea and V the volume, _ = ya — avy’, v and y 
denoting derivatives with respect to. Hence by Guldins theorem, 


1 By ‘‘exists’’ is meant that ie approaches a determinate value, whether finite 
or infinite. ? 
y 2 When we are unable to distinguish the in- 
terior from the exterior, the attraction of part 
of the matter as given by A (see b), may 
become negative ; e.g., the contribution of the 
loop, 1, in fig.5 ; a state of affairs that may be 
realized physically in statical electricity, at 
least in the imagination. In such a case, it is 
evident that we can make the attraction as 
great as we please by placing the positive elec- 
tricity in sufficiently great quantities near the 
(Fi S. 5) negative particle, and enough negative electri- 
city to satisfy the isoperimetric condition at 
such a great distance to render its attraction infinitesimal. More explicitly, 
suppose that a charge, à and ¢ are to satisfy the isoperimetric condition, 
@ 
& ee =%>O0. Select any positive r, r, > — and any À, R > 2r; an « = 2r° A, 
ad 
and e, = 272 A—w. Then charges, à, and — e on two spheres whose centres are 
collinear with the attracted particles and at distances, r, R, from it, exert an 
attraction 

F ey eh 2r? A ar? A = 2 14 2) re 
Er NL ee 1 ig LE PTS A. 
ro By 72 PR? Le? 
Now, À is any arbitrary quantity. Hence F can be made as great as we please. 
The absence of such loops is therefore an essential condition for the existence of 
a maximum. 
