590 PLATEAU ON THE PHENOMENA OF A FREE LIQUID MASS 



thus led to a complicated differential equation, which only ap- 

 pears susceptible of integration in particular cases. Yet the 

 equation (3.) will be useful to us in the above simple 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 otlier. 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 p', corresponding to the two sections thus determined, will 



be such that the quantity - + — will be equal to the quantity 



p- + -jy,. Hence the first of these two quantities may be substi- 

 tuted for the second ; and consequently, the equation of equili- 

 brium, in its most general expression, will be 



J+?=C' W 



in which 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 sec- 

 tions at the point of the surface under consideration. This being 

 admitted, if we imagine that the system of the two planes occu- 

 pies successively difierent positions in turning around the same 



1 c . 1 . 1 1 1 1 , 1 e 

 normal, a sum or curvatures —-'r—,,-r, + -m> "vH ? o^c. will 



? _ p r r P P 



correspond to each of these positions; and, according to the 

 property noticed in the pi'eceding paragraph, all these sums 

 will have the same value. Consequently, if we add them toge- 

 ther, and let n denote the number of positions of the system 

 of the two planes, the total sum will be equal to n times the 



value of one of the partial sums, or to n ( — t" T ) • ^ow this 



total sum is that of all the curvatures — , -r, -j-,, -m, &c. in number 



P P P P 

 2n, corresponding to all the sections determined by the two 



