a all 
344 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
its meridian section or vertical projection, by Fig. 7, a m and 4 m being the sec- 
tions or projections of the disks. It will be stated hereafter for what reasons 
we have suggested the use of a cylinder rather thangof disks or rings. 
§ 6. The figure which we have thus obtained, and in which the meridian line 
stops at the points a and 6 where it touches the cylinder (Fig. 6) or meets the 
borders of the disks (Fig. 7,) evidently constitutes but a portion of the complete 
figure of equilibrium. Let us attempt then to follow the meridian line, start- 
ing from these same points a and 4 where its elements are parallel to the axis. 
It is easy to show that the points a@ and 6 are not points of inflexion. At 
such points the radius of curvature is either null or infinite; but since, in our 
meridian lines, there can be no question of a radius of curvature null, which 
would render the first member of the equation of equilibrium infinite, it would 
be necessary to suppose this radius infinite at the points which we are consider- 
ing, and the equation would there be reduced to ~ =O. Now, the points candd 
(Fig. 6) are really points of inflexion of this kind, as the aspect of the figure 
shows, in so much that the equation of equilibrium is there necessarily reduced 
1 
tor =G; the perpendicular N should then, at the points @ and 4, have the 
same length as at the points ¢ and d, which is evidently not the case; for, in 
the first place, the points ¢ and d are more remote from the axis than the points 
a and 6, and, moreover, the perpendiculars which proceed from the former are 
oblique to the axis, while those which correspond to the latter are perpendicular 
to it. 
Beyond the points @ and 4, then, the curve begins by preserving a curvature 
having the same direction as before, that is,a curvature concave towards the ex- 
terior (Fig. 8.) Now, let us suppose that in the prolongation starting from a, 
for instance, this curvature should continue either augmenting or diminishing 
less than it diminishes on the other side of a; we can always take on the pro- 
longation in question a portion a m so small that at each point the curvature 
shall be stronger than at the corresponding points of a portion a m of the same 
length taken on the first part of the curve. By virtue of the greater curvature 
of all the points of the are a m, the point m is necessarily more remote from the 
axis than the point m, and, moreover, the perpendicular m 7 which proceeds from 
the former is more oblique to the axis than the perpendicular n s which proceeds 
from the second; the perpendicular at m is, for this double reason, greater than 
the perpendicular at ~. On the other hand, conformably with the same hypothe- 
sis relative to curves, the radius of curvature at m is smaller than atm. ‘Thence 
it results that.in passing from the point 2 to the point m, the first term of the 
quantity MON will increase and the secund diminish. Now, in the parts of 
the curve which we are considering, the radius of curvature and the perpendic- 
ular are opposite to one another, and have consequently contrary signs, so that 
Bg Seg pe ; ; : . 
the quantity MN constitutes a difference; if, then, one of the terms of this 
