108 TRANSFORMATION OF SURFACES BY BENDING. 



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 con- 

 secutive 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 

 consideration 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*z d*z d\ I* 

 chi? dy* dxdy\ 



dy 



The definition of conjugate systems of curves may be rendered independent 

 of the reasoning formerly employed by the following modification. 



