358 The Rev. J. H. Jellett on the Properties of Inextenslhle Surfaces. 



Similarly we shall have 



V =A'x + B'. 



w = A"x+B". 



Hence it is easy to see that if./, y\ z' be the co-ordinates of the new position 

 of the point x, y, z, we shall have 



x' = a.v + b, 

 y' = a'x + h', 

 z' = a"x + b'\ 



where a, a\ a", b, b', b", are constant for the same rectilinear section. From 

 these equations it is plain that the locus of the points x', y', z' is still a right 

 line. 



III. Concavo-convex surfaces, or tliose in ichich 



rt -s'- <0. 



It is a well-known property of surfaces of this class that at each point of the 

 surface there are two real directions satisfying the condition 



r cos- a + 2s cos a cos j3 + tcos- /3 — ; ( W) 



an equation which expresses the geometrical fact, that the normal section which 

 passes through either of these directions will have at that point an infinite 

 radius of curvature. We may therefore conceive the entire surface to be 

 crossed by two series of curves, such that a tangent drawn to either of them at 

 any point shall possess this geometrical property. These curves we shall de- 

 nominate (for a reason which will appear subsequently) curves of flexure. "We 

 shall consider separately (as before for developable surfaces) the two different 

 cases which arise, according as the fixed curve is or is not a curve of flexure. 



1. When the fixed curve is a curve of flexure it is evident, as in the case of 

 developable surfaces, that the equation 



(r + 2sm + tm-) -r-r - 

 ^ dxay 



becomes identically true without supposing 



ih'dy 



