342 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
6 
while, as regards the other, it is directed to the exterior, and is consequently 
: 1 ec : t 
negative. Now, the equation aty=e cannot comprise this change of sign of 
the perpendicular N in passing from one branch to the other, for it would re- 
quire that at the point of retrogression this perpendicular should be null or in- 
finite; and in the present case the perpendicular in question is evidently finite, 
since the tangent is not perpendicular to the axis, and the point of retrogression 
cannot be upon the latter. 
Second case —lIf the point of retrogression be of the second kind—that is to 
say, if the two branches which meet therein are situated on the same side of the 
common tangent—we see that, for one of these branches, the perpendicular and 
the radius of curvature are both positive, while for the other they are both 
Me: ° sa ag ! ; 
negative; the quantity MON would then change the sign in passing from one 
to the other, and thus would not be the same through the whole extent of the 
liquid figure. 
If the point of retrogression is of the first kind—that is, if the two branches 
are situated on the two opposite sides of the common tangent—the radius of cur- 
vature, we know, is there null or infinite, but a radius of curvature null would 
render infinite thé quantity ale so that we have to examine only the hypoth- 
* 
esis of a radius of infinite curvature. Since, then, from the direction of the tan- 
gent, the perpendicular is also infinite at the point which we are considering, 
Pi cel ia : 4 
the quantity tay would be reduced to zero at the same point; it would there- 
fore be necessary, for equilibrium, that this quantity should also be null at all 
other points of the meridian line. Now, this is impossible, since, when we depart 
from the point of retrogression, the radius of the curvature and the perpendicu- 
lar assume, on each of the branches respectively, values finite and of the same 
sign. 
Third case.—If the point of retrogression is of the second kind, the radius , 
of curvature has opposite signs on the two branches, and consequently must be 
either null or infinite at the point in question; but, as has been already shown, 
we need not occupy ourselves with the hypothesis of a radius of curvature null; 
there remains, therefore, that of a radius of curvature infinite. Now, the perpen- 
dicular at the same point being likewise infinite, equilibrium requires, as above 
that the quantity ath should be null for all the points of the meridional line. 
Here, at first glance, the thing seems possible, since, near the point of retrogres- 
sion, the radius of curvature and the perpendicular, on each branch considered 
separately, are of contrary signs, but we shall presently see that this possibility 
is but apparent. 
If the point of retrogression be of the first kind, the radius of curvature is 
there necessarily null or infinite, as has been already shown; and since we must 
; = , Lins , 
reject the radii of curvature null, the quantity vee is again equal to zero at 
the point in question, and must be so likewise at all other points; which appears 
possible as in the former ease, and for the same reason. But in order that at 
all points of the meridian line the quantity = ae should be null, it is evidently — 
necessary that in each of these points the radius of curvature must be equal 
and opposite to the perpendicular. Now geometers are aware that one curve 
alone possesses this property, and that that curve is the ecatena, (chainette,) 
which has no point of retrogression. 
