346 The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 



These equations may be put under a somewhat simpler form, by assuming 



u = lx ■\- pdz, 



V = ly + qlz., 



10 = Iz. 



Making these substitutions, we find 



du 



icr = 0, 



ax 



(III dv n A m-\ 



-^+^-2ivs = 0, . (C) 



dy dx 



dv ^ 



ivt = 0\ 



dy 



where r, .?, f, are used in their ordinary sense to denote the differential coeffi- 

 cients 



d-z d-z d'z _ 



dx'^ ' dxdy ' dy- ' 



derived from the equation of the surface. Any one of the quantities u, v, w, may 

 be determined by means of a differential equation of the second order. Thus, 

 for example, eliminating w between the equations (C), we find, 



\ du _ 1 fdu dv\ _ 1 dv 

 r dx 2s \dy dx) I dy ' 

 Hence, 



dv _ 25 du du 



dx ~ r dx dy'' 



dv t du 

 dy r dx' 



Differentiating the first of these equations with respect to ?/, and the second 

 with respect to x, and subtracting, we find easily, 



d-u d^u d^u _\ d {rt — S-) du ,j.-. 



dy^ dxdy dx- ~ r dx dx' 



and similarly for v, 



d^v „ d'v d-v 1 d (rt — S-) dv ,^^ 



7- ^2 - 2s ^-j- + < ^-, = - -■!— ^ ^ -J-. (E) 



f/y dxdy dx- t dy dy 



