4>66 Mr MAXWELL, ON THE 



We are now able to determine whether any system of lines drawn on a given surface is a 

 system of instantaneous or permanent lines of bending. 



We are also able, by the method of Article (8), to deduce from two consecutive forms of 

 a surface, the lines of bending about which the transformation must have taken place. 



If our analytical methods were sufficiently powerful, we might apply our results to the 

 determination of such systems of lines on any known surface, but the necessary calculations 

 even in the simplest cases are so complicated, that, even if useful results were obtained, they 

 would be out of place in a paper of this kind, which is intended to afford the means of forming 

 distinct conceptions rather than to exhibit the results of mathematical labour. 



18. On the application of the ordinary methods of analytical geometry to the considera- 

 tion of lines of bending. 



It may be interesting to those who may hesitate to accept results derived from the 

 consideration of a polyhedron, when applied to a curved surface, to inquire whether the same 

 results may not be obtained by some independent method. 



As the following method involves only those operations which are most familiar to the 

 analyst, it will be sufficient to give the rough outline, which may be filled up at pleasure. 



The proof of the invariability of the specific curvature may be taken from any of the 

 memoirs above referred to, and its value in terms of the equation of the surface will be found 

 in the memoir of Gauss. 



Let the equation to the surface be put under the form 



then the value of the specific curvature is 



d 2 z d?z a?z 

 das 1 dy° dxdy 



P- 



^ 



dz 

 don 



dz 

 dy 



The definition of conjugate systems of curves may be rendered independent of the 

 reasoning formerly employed by the following modification. 



Let a tangent plane move along any line of the first system, then if the line of ultimate 

 intersection of this plane with itself be always a tangent to some line of the second system, the 

 second system is said to be conjugate to the first. 



It is easy to show that the first system is also conjugate to the second. 



Let the system of curves be projected on the plane of xy, and at the point («, y) let a be 

 the angle which a projected curve of the first system makes with the axis of x, and 3 the 

 angle which the projected curve of the second system which intersects it at that point makes 

 with the same axis. Then the condition of the systems being conjugate will be found to be 



d*% „ d'z . • ^ <fjr ■. . _ „ 



-— cos a cos 8 + - — r sln ( a + P) + T~ 9 Sln a sin jB = ; 

 dx* r dxdy ^' dy* 



a and 3 being known as functions of * and y, we may determine the nature of the curves 

 projected on the plane of xy. 



