352 The Rev. J. H. Jellett on the Properties 0/ Inextensible Surfaces. 



severally the three great classes into whicli surfaces are divided with respect 

 to their curvature, namely : 



1. Surfaces whose principal curvatures are similar, or those in which 



rt — s' > 0. 



2. Developable surfaces, in which 



rt- s- = 0. 



3. Surfaces whose principal curvatures are dissimilar, or those in which 



rt-s" < 0. 



I. Surfaces whose principal curvatures are similar. — In this case it is plain 



that the equation 



r + 2sm + tm'^ = 



is impossible, whatever be the value of m. Therefore the equations (N) can 

 only be satisfied by making 



dhi ^ d'^v 



(O) 



which must hold throughout the fixed curve. Again, differentiating equations 

 (D) and (E) (which are true generally) with regard to x, and rejecting diffe- 

 rential coefficients of the firbt and second order, which vanish for the fixed 

 curve, we have 



d^u d^u d^u _ 



dxdy^ dx'-dy dx^~ ' (Pj 



d^v „ d^v d'u 



— 2s , , , + < J— 3 = 0, 



dxdy' dx'dy dx 



which must hold throua;hout the fixed curve. 



