The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 357 



+2.$+^$;=o, 



(S) 



, du (III , du . ,rr. 



rd.-r + t-i'-.d.-r-^O. (T) 



dy ax dx 



Let p^Q 



be the equation of the given surface ; then 



r = Q's, s = Q't, 



where n' —^^. 



dq 



Substituting these values in equation (S), we find easily 



dx 



Now since equation ( S) represents a rectilinear section of the surface, it is evi- 

 dent that in this equation Q' must be constant. Hence it becomes 



y + Q!x — const. 



dv 

 Again, substituting for ?■, ;, and -j^ in equation (T), and integrating, we find 



„, du du 



Q -. J- = const. 



dy dx 



Then the integral of equation (E), which may readily be obtained in the ordi- 

 nary way, will be 



u =. xf{q) -f F{q). (V) 



The following general property of the motion may be deduced from this 

 equation : 



The rectilinear sections of the surface are rigid* 



For, since in a developable surface q is constant for the same rectilinear sec- 

 tion, the value of u for such a section will be 



u- Ax + B, 



A, B being constants. 



* It is easily seen, however, that these sections may all bend at the arke de rebroussement. 



3 A 2 



