The Rev. J. II. Jellett on the Properties of Inextensible Swfacea. 353 

 Differentiating equations (M) as before, and eliminating 



(Pu dhi 



from equations (P), we have 



(r + 25m + tnr) j-j--^ = 0, 



' ^ (Q) 



(Pv 

 (r + 2sm + tm-) ■, , , = 0. 

 a.my' 



Hence it is easy to infer, as before, that all the differential coefficients of the 

 third order vanish for points of the surface situated on the fixed curve ; and a 

 very slight examination will show that by proceeding in the same manner we 

 shall find that all the differential coefficients of u, of all orders, vanish for the 

 limiting curve. Now if m be a function of the same form throughout the sur- 

 face, it is plain that these conditions can only be satisfied by the supposition 

 that u vanishes at every point. The same conclusion will hold if u change 

 its form. For if u be supposed to have the same form for all points between 

 the limiting curve and any other curve drawn arbitrarily, it is plain, from what 

 has been said, that its value can be no other than zero. Hence for all points of 

 the second curve 



M = 0. 



Now it is evident that the same reasoning which was before applied to the 

 limiting curve is equally applicable to this second curve, and so on for any 

 number of curves bounding those parts of the surface for which the form of u 

 is the same. It appears, therefore, from the foregoing reasoning, that we 

 must have throughout the entire surface 



By precisely similar reasoning it may be shown that we have throughout the 



entire surface 



v = 0; 



and on referring to equations (C), it will be seen that it follows at once from 



these equations that 



ia = 0. 



