WITHDRAWN FROM THE ACTION OF GRAVITY. 229 



The analytical expression of this condition is directly deducible from the. 

 general value of the pressure given in the preceding paragraph ; we only require 

 to equalize this value to a constant, and, as the quantities P and A are them- 

 selves constant, it is in fact sufficient to make 



i+w^"" ,<^' 



the quantity C being constant for the same figure of equilibrium. 



This equation is the same as those which are given by geometricians for ca- 

 pillary surfaces, when, in the latter equations', the quantity representing gravity 

 is supposed to be O. 



R and R' may be replaced by their analytical values; ^ve are thus led to a 

 complicated differential equation, which only appears susceptible of integration 

 in particular cases. Yet the eqiiation (3) will be useful to us in the above sim- 

 ple form. Now we know that the normal plane sections which correspond to 

 the greatest and the least curvature at the same point of any surface form a 

 right angle with each other. Geometricians have shown, moreover, that if 

 any two other rectangular planes be made to pass through the same normal, 

 the radii of curvature, p and /?', corresponding to the two sections thus deter- 

 mined, will be such that the quantity 1 — ; will be equal to the quantity 



-^ + TT",. Hence the first of these two quantities may be substituted for the 



second ; and, consequently, the equation of equilibrium, in its most general 

 expression, will be 



- + -,= 0, ■ (4.) 



in Avhich equation p and p' denote the radii of curvature of any two rectangular 

 sections passing through the same normal. 



6. These geometric properties lead to another signification of the equation 

 (4.) We know that unity divided by the radius of curvature corresponding to' 

 any point of a curve is the measure of the curvature at this point. The quantity 



— + — represents, then, the sum of the curvatures of two normal rectangular 

 P f' 



sections at the point of the surface under consideration. This being admitted, 

 if we imagine that the system of the two planes occupies successively 

 different positions in turning around the same normal, a sum of curvatures 



— -I , 1 , -^ -\ , &;c., will correspond to each of these positions ; 



p p' p" p'" p^^ p^ 



and, according to the property noticed in the preceding paragraph, all these 

 sums will have the same value. Consequently, if we add them together, 

 and let n denote the number of positions of the system of the two planes, 

 the total sum Avill be equal to 7i times the value of one of the partial sums, 



or to « B ' 1 B. Now, this total sum is that of all the curvatures 



— , — , — , — , &:c., iu number 2)1, correspondiuo' to all the sections determined 



/ )r III ' i o 



P p' p" p'" 



by the two planes. If, then, we divide the above equivalent quantity by 2n, 



the result — ( — -\ ; 1 will represent the mean of all these curvatures. Now, 



2\/> f' J 



as this result is independent of the value of n, or of the number of positions 

 occupied by the system of the two planes, it will be equally true if we suppose 



