514 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



where dS is an element of area of the middle-surface and ds an element of arc of the 

 edge. 



Observing that by equations (15) 8n s = 0, and integrating with respect to z from h 

 to h, we get the equation 



- j](X Su + Y Sv + Z 8w) dS - j](M 8/ 8 - L 8m 3 ) rfS 



- |( A 8u + B 8v + C Siv) ds - f (V S1 3 U Sm 3 ) ds 



j d$ + 2nh 8W 2 



9, (10) 



which contains in itself all the equations and conditions of the problem. 



All the double integrals which occur in this equation can be expressed, partly as 

 surface-integrals over the middle-surface, and partly as line-integrals round the edge, 

 by means of the theorem, 



A, ..... (20) 



where the first integration extends to all values of (a, ft) which correspond to points 

 on a surface having s for an edge, and X, y. are the cosines of the angles which the 

 normal to the edge drawn on the surface and produced outwards makes with the 

 directions of the lines ft = const., a = const, at the edge. 



To prove this theorem,* let a line of curvature a = const, meet the edge in an even 

 number of points, and let X 1} X 2 , ... be the values of X at these points, then 



80 



The partial integrations will be effected by means of the relations 



a / aa^ ax 



ax 



* (7/. MAXWELL, 'Electricity and Maj^netism,' Art. 21. This theorem is otherwise proved by A RON. 



