108 On an Elementary Proof of a Theorem of Lagrange's. 



the tension along a length ds is 



\ds. 



Around the circumference of the ellipse the tension of any ele- 

 ment of the arc acts in the tangent-plane and perpendicular to the 

 arc. The direction-cosines of the normal at the point xyz are 



and the direction-cosines of the element of the arc of the ellipse 

 are 



doc dy 



ds' ds' 



Hence, since these lines are perpendicular, the cosine of the 

 angle which a line that is perpendicular to both of them makes 

 with the axis of z is 



y dx x dy 



K 2 ds Rj ds 



The tension of the element ds is to be resolved along the axis of 

 z by multiplying \ds by this quantity, the product being 



Now, according to Lagrange's notion of the nature of such a 

 membrane, X is constant around this ellipse. 



Let then x=asmcf>, y = b cos <f>. 



The entire tension of the surface along the ellipse resolved along 

 the axis of z is 



4\« b \ (g- cos 2 ^ + 4- sin 2 <p) . d<f> 



=7rXa Ks; + i) ' 



but this must be equal to the normal pressure on the ellipse, 

 which is 



irabVj 

 hence 



x (a; + s-> p - Q- ED - 



Royal College of Science for Ireland, 

 December 29, 1869. 



