oOfi 



MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



parallel to ft = const., v to a = const., and w along the normal outwards, and let 3/3n s 

 denote differentiation along the normal. 



The square of the length of a line joining two near points of the surface before 

 strain is 



and the square of the length of the line joining the same two points after strain is 



i 



o 



neglecting small quantities of a higher order. But this same square of the new length 

 of the line is 



where 8 (l//^), 8(l/h 2 ) are the increments of l/h lt l/h. 2 produced by strain, so that 



l 



* 



S ( , = A.M^- 



and similarly for & 2 , also 



and so for d (h. z v). 



In the two expressions for the square of the new length of the line we may equate 

 coefficients of (dctf, (df$f, and (da. d/B), and omit powers of u, v, w, or their 

 differential coefficients above the first ; thus 



a /i 



w 



a . 



where we have written 



in accordance with LAMP'S result (' Lecons sur les Coordonnees Curvilignes,' p. 51), 

 viz., pj, p. 2 are the principal radii of curvature of the middle-surface before strain, 

 reckoned inwards. 



