The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 369 



of the quantities dlx\ dcf, lUz', and since this is true for all directions of the 

 arc ds, it is plain that the coefficients of each of the quantities 



dx^ dy^ dx dy dx dn dy dn dir 

 d7'' 17'' d?dP' d7'd^' dPd^" 57^' 



must be indefinitely small as compared with 8.r, iy, 8z. "We shall, in the first 

 place, consider the coefficients of the first three of these quantities. 



If we neglect, as before, quantities of the second order, we may evidently 

 substitute in these coefficients Zx, cy, Iz, for E.r', 8y', iz'. We shall have then 



dZx dhz 



dx " dx ~ ' 



dix dly dlz doz 



dy dx ^ dy dx ~ ' 



dhy dhz 



be satisfied, where a, J, c are functions of x and y of the same order of magni- 

 tude as hx, iy, cz. 



Transforming these equations as in p. 346, we find 



du 



tcr = la, 



dx 



dv 



Wt= w. 



dy 



Now it is well known, that such a system of equations may always be satis- 

 fied by the values 



u = u' + iui , 



V = v' + ivi , 



w = w' + iw, ; 



where u', v', w\ satisfy the equations 



