[ 107 ] 



XIII. Note on an Elementary Proof of a Theorem of Lagrange's. 

 By Professor Robert Stawell Ball, A.M.* 



" [F a membrane, extensible or inextensible, be in equilibrium 

 A under the influence of forces at every point normal to the 



surface, then the normal pressure at any point is equal to the 



tension of the surface at that point, multiplied into the sum of 



the reciprocals of the principal radii of curvature/' 



This theorem is at present of great interest, in consequence of 



its very beautiful application in the recent discoveries of Professor 



Haughton. 



Taking the origin on the surface, the axis of z for the normal, 



and the axes of x and y tangents to the principal directions of 



curvature, the equation of the surface is 



2z--~- y - 

 R, R 2 



4- terms of the third and higher orders =0. 



But as we shall only deal with points in the immediate vicinity 

 of the origin, we may neglect all but the three first terms, and 

 thus 



Now, let a plane parallel to the tangent-plane, and separated 

 from it by an infinitely small interval^, be drawn; this will cut 

 the surface in an ellipse defined by the equations 



z=p, 



a** b*" 1 ' 

 if 



a 2 =2R 1 p and6 2 =2R. 2 p. 



Let us fix our attention on the small piece of the surface thus 

 cut off. It may be considered plane, because the greatest devia- 

 tion of any part of it from the plane is of the second order of 

 small quantities. Hence, if P be the normal pressure per unit of 

 area to which the surface is at this point exposed, the total pres- 

 sure on the piece bounded by the ellipse is 



7T«Z>P. 



This pressure is counteracted by the tension of the surface which 

 acts along the margin of the ellipse. 



Let \ be the tension of the membrane per unit of length, then 



* Communicated by the Author. 



