360 The Rev. J. H. Jellett on the Properties of Inextensilk Surfaces. 



No one, therefore, of the functions u, v, w, can change its form, except in pass- 

 ing across a curve of flexure. Hence the proposition is evident. 



(II.) Let AB, AC be two arcs of curves of 

 flexure commencing at the same point A. Through 

 B, C draw the curves of flexure BD, CD, meeting 

 in D. Then if AB, AC be fixed, the entire qua- 

 drilateral ABDC is fixed also. 



The truth of this theorem is nearly evident 

 from the theory of partial differential equations, 

 combined with the principle laid down in (I.), but it may be strictly proved as 

 follows : 



Since w, v, w, can only change their forms in passing a curve of flexure, we 

 may suppose them to retain the same form throughout the entire of the quadri- 

 lateral AiifZiCi, formed by drawing the curves of flexure b^d., CicL 



Let e = c, e' = c', 



be the equations of the two series of curves of flexure. Then, since the func- 

 tions e, 0' satisfy the differential equations 



dy- 



de" 



dx dy dx^ ' 



do'de' do" 



dy^ dx dy dx' 



if the independent variables x, y, be changed into 6, 6', the equation 



d'w 



_ d-iv „ d-w ^ 



dy' dxdy dx' 







will (as is well known) assume the form 



d'w p dw ^ dio _ 



dede' ^ W^^de'' ' 



P, Q, being functions of 0, 6'. Now since iv vanishes for the curve 



e = c, 



it is plain that we must have throughout this curve, 

 dw 



(X) 



— = 



de' ~ ' de" ' 



^ = 0, &c., 



