340 FIGURES OF EQUILIBRIUM OF A LIQUID MASS . 
points ; this quantity would, therefore, not be constant in the entire line of the 
curve as the equation of equilibrium requires. 
Let us imagine, now, that the liquid fulfils the condition just laid down, and 
proceed to consider, at its departure from the axis, an are of the meridian line. 
As, by the hypothesis, this line.is neither straight nor cireular, the curvature of 
the are will vary from one point to another; it will commence, consequently, 
either by a process of augmentation or diminution, and we may take an are so 
small that the curvature shall go on constantly augmenting or constantly dimin- 
ishing from the point situated on the axis, quite to the other extremity. Let 
us suppose, first, the curvature continually increasing, and let a 6 d (Fig. 1) be 
the arc in question. At the point @ the perpendicular is coincident with the 
axis, and, in proportion as it leaves that point, forms with the axis a progress- 
ively greater angle; but we will so limit the length of the are that, from a to d, 
this angle shall not cease to be an acute one. Through the two points, a and d, 
let us describe an are of a circle, a c d, which shall have its centre on the axis, 
and which, consequently, meets this axis perpendicularly. 
Fig 2 Fiy3 Pye Fig 
Ba f 
a4 \ 
“y 
al PA vA 
Since the are a 4 d, whose curvature constantly increases, departs from @ in 
the same direction with the are of a circle, and, after being separated from it, 
rejoins it at d, it is evident that its curvature must, at first, be less than that of 
this second are, and afterwards become greater, so that at the point d the radius 
of curvature of the are a 4 d is smaller than the radius of the are of the circle. 
But from the common initial direction of the two arcs, and from this relative 
progression of the curvature of the are @ & d, it necessarily results that this 
last is, as the figure shows, exterior to the other, and that at the point d it must 
cut and not be a tangent to it; if, then, at this point d, we draw the perpen- 
dicular d f to the are of the curve and the radius d g of the are of the circle, 
the former will be less oblique to the axis than the latter, and will conse- 
quently be shorter. Thus, at the point d, the two quantities M and N will be 
both less than the radius of the are of the circle. Let us take now, in the part 
of the are a b d, where the curvature is less than that of the arc of the circle, 
any point m, and let us take, on the second of these ares, a point 2, so that the 
portion a z shall be equal in length to the portion a m. Under these condi- 
tions, the point m will be evidently more remote from the axis than the point x, 
entirely annulled. We may remark, in passing, that this result is independent of the con- 
dition m<2, so that it is true as well for a radius of curvature, finite or infinite, at the point 
situated on the axis, as for a radius of curvature null, which should be the case according 
to what has been seen above. Now, if at this same point the radius of curvature is null, the 
a 
W 
becomes at the same time infinite, their difference becomes also infinite, which is what was 
required to be demonstrated. 
two quantities N and both assume, indeed, an infinite value; but, since their ratio 
