1 ftlu 

 R 



The Rev. J. II. Jellett on the Properties of Inextensihle Surfaces. 363 



If a surface have at each point three or more inextensihle directions, it is ichoUy 

 inextensible. 



A surface may have at each point one or two inextensihle directions, without 

 being whoUy inextensible. 



Suppose that the given surface has at each point two inextensible curves 

 included in the equation 



Edx' + "iSd.Tdy + Tdy'' = 0, 



or B cos- a + 2S cos a COS /3 + Z'cos- /3 = 0. 



Then, as this equation must be identical with 



fdn \ , fdu dv ^ \ ^ fdv \ , ^ „ 



I -r- — wr 1 COS"^ a + ( -^ + -T 2ll-S 1 cos a COS /5 + ( T M'M cos^ ^ = 0, 



we shall have 



(du \ 1 fdu dv ^ \ \ fdv \ 



These two equations contain the entire theory of tlie surfaces under con- 

 sideration. 



Suppose, fur example, that the surface is one of dissimilar curvatures, and 

 that its curves of flexure are inextensible. We have then 



R — r, S — s, T = t, 



and the equations (Z) become 



rdx''2s\dy^'d:r)~ld^' ^^ 



being identical with the equations which are found by eliminating w between 

 the general equations (C), p. 346. The displacement iv remains indeterminate. 

 From these considerations it is easy to deduce the following theorem : — 



If tlte curves of flexure traced upon a surface with dissimilar curvatures be in- 

 extensible, the most yeneral displacement of tvhich the surface is capable may be 

 found by supposing it first to move as an inextensible surface, and then to receive at 

 each point a normal displacement of arbitrary magnitude. 



VOL. XXII. 3 B 



