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* WITHDRAWN FROM THE ACTION OF GRAVITY. 339 
Let us conceive a figure of equilibrium of revolution not being either a 
sphere or a plane, and whose meridian line meets the axis. I maintain, in the 
first place, that this line can attain the axis only perpendicularly. In effect, if 
it intersected it obliquely, or if it were a tangent to it, the perpendicular would 
be null at the point of intersection or of contact, and the quantity se would 
become infinite at that point,* while it would be of finite value at neighboring 
* There is one case, however, in which this reasoning would seem not to be applicable. 
We may conceive a curve such that, at the point where it meets the axis, the radius of curva- 
ture would be null, and that in the neighborhood of this point the radius of curvature and 
the perpendicular would be of opposite signs; then the quantity a +5 would constitute a 
difference, of which both terms would at once become infinite at the point situated on the 
axis, and it is not apparent, at the first glance, that this difference might not remain finite. 
We have to demonstrate, therefore, that the thing is in§possible if the curve does not meet 
the axis normally. For that purpose, but only in this case, we shall be obliged to make use 
of the known expressions of the radius of curvature and of the perpendicular in functions of © 
differential coefficients. 
If we take the axis of revolution as axis of the abscissee, we shall have p and gq, respect- 
ively the ditferential coefficients of the first and second order of y relatively to xz: 
1 2 3 
q 
N=y (1+ 2)? Deli erie alolsl Sol osese sige eke Sele Uaieing 1 (2) 
whence we deduce, for the relation of the two terms of the first member of the equation of 
equilibrium : a 
¢ salt 
N 1+ 7? 
—=— Mes eR 5 ales Ph gent ais A uh ae Lu sen ps dl (Gh 
fal ae (3) 
M 
Now, let y=/f (x) be the equation of the meridian line. Taking as origin of the co-ordinates 
the point where this line meets the axis, so that for co we have yo, we can then sup- 
pose the function f(x) developed in a series of ascending and positive powers of z; and if 
we assume that the curve meets the axis under an angle other than a right angle, which re- 
quires that, for =o, the first differential coefficient should be finite or null, it will be neces- 
sary that the exponent of zx, in the first term of the series, should be unity. Let us remark 
here that, having only to consider the curve at the point where it reaches the axis and at 
points very near, we may always consider z extremely small, so that, in relation to this por- 
tion of the curve, our series will be necessarily convergent. Let us say, then: 
ee le ce eee ey a enn See are oe eae ete oe (4) 
an equation in which the exponents m m.... are positive and greater than unity. Conse- 
quently we shall have: 
ae] MO ED ee eG a eee eiocls Jee ICL ok OR 
mn (mL) Oe te entra aN PS a ek 
The first of these expressions, when we make therein xo, is reduced to p=a, so that 
the curve meets the axis under a finite angle, but other than a right angle, or under an angle 
null, according as we suppose the constant a finite or equal to zero. Then, if we assume 
that at the point situated on the axis the radius of curvature is null, we see, by formula (1, ) 
that in this same point g must be infinite, and, in virtue of the second of the above expres- 
sions; this condition will be satisfied if the first, at least, of the exponents mn-.... be less 
than 2. 
Let us now introduce into the formula (3) these same expressions of p and qg and that 
of y. There will result: 
1 
‘ih A Mi UN el a le ale aa 
1 (m(m—1) b2™—? fn (n—1) co®*— $f -.--) (ae Foe caf...) 
M 
and we readily see that, for zo, this ratio becomes infinite. Then, in effect, since the 
quantities m m.... are all greater than unity, on the one hand the numerator is reduced to 
1-- a?— that is to say, to a finite quantity; and, on the other hand, the denominator, of which 
gine term of the smaller exponent, after the requisite multiplication, is m(m—1) abz™—, is 
