368 The Rev. J. H. Jellett on the Properties of Inextensihle Surfaces. 



Differentiating these equations, and rejecting ncla, nd^, Jidy, on account of the 

 small quantity n, which is of the same order as the thickness of the membrane, 

 Ave have 



d.v' = (Ir + adn, 



ciy = dy + /3(/«, (H') 



dz' = dz + 7(/h. 



If now we represent by ds' an arc of a curve traced in the substance of the 

 membrane, we shall have, as before, 



ds'lds' = dx'dlx' + dy'dZy' + dz'dl:f. (I' ) 



But if we regard Ix', ly\ Iz' as functions of the three variables x, y, 71, we 

 shall have 



, , dcx' , dix' , dix' , 



dZy'^'^dx+'^dy + '-^dn, (K') 



•^ dx dy •^ dn 



dlz' , dlz' , dlz' , 

 acz = —j — dx H — ;— dy + —,— dn. 

 dx dy dn 



Substituting in (!') the values (H') and (K')i ^^i^ putting for dz its value 

 pdx + qdy, we have 



dos' _ f dlx" dlz'\ d£_ /(%' dlz\ dy^ 

 ds ~ \ dx ^ ^' dx ) (77- ^ \1^ ^ ^ ~d^) d7^ 



/dha/ dZj/_ dis^ dh3'\dxdy 

 \ dy dx dy ' dx J ds' ds' 



fdlx' dhz' dKv' ^dZy' dlz\dx dn ,^ ,, 



fdiy' dhz' dlx' dly' dlz'\dy dn 

 / dlx' ^dly' dhz'\dn^ 



Now since dcs' is by hypothesis indefinitely small, as compared with any one 



