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



Abi, bibi, bibi, &c., formed by drawing curves of flexure through 6,, 62, &c. 

 Hence, and from p. 359, it is evident that zcmust vanish throughout the entire 

 of each of these quadrilaterals. But if 



IV = a 



for the quadrilaterals Ai,, b^bn, it follows from Theorem 11. p. 360, that w must 

 vanish for the quadrilateral bid ', and by pursuing the same method we shall 

 easily see that we must have 



?i' = 



for each of the quadrilaterals into which ACBD is divided. Hence the truth 

 of the proposition is evident. This proposition may be expressed by saying 

 that 



If an arc of a curve traced upon an inextensible surface be rendei'ed fixed or 

 rigid, the entire of the quadrilateral, formed by drawing the two curves of flexure 

 through each extremity of the cui-ve, becomes fixed or rigid also. 



6. We shall now proceed to consider the case of surfaces which, without 

 being wholly inextensible, have at each point one or more inextensible di- 

 rections. 



Reverting to the discussion of p. 345, and making 



dx dy 



-^ = cos a, -^ — cos B, 



as as 



~ = ( ^ — 2cr ] cos-a + ( -T- + ;i 2iTO ) cos a cos P + [-j tvt) cos'^. (Y) 



we find easily 



ids (du \ „ (du 



From this equation it is plain that, unless the coefficients of 



cos^ a, cos a cos j3, cos^ /3, 



vanish separately, there can be, for each law of displacement, but two values of 



cos a 

 cos/3' 

 which will satisfy the equation 



Us = 0. 



If these coefiicients vanish separately, ids will vanish for every direction round 

 the point. Hence it is easy to infer the following theorems : — 



