The Rev. J. H. Jellett on the Properties of Inextensible Surfaces. 365 



cds dcx „ dly , ^ dcz , 



ds dx dy dz (B ) 



(dlz dly\ ^ (dcx dlz\ Idni dcx\ 



Now if the body be inextensible, we must have for all values of a, j3, 7, 



Ids = 0. 

 Hence we have the six equations, 



^_A l^_n ^-n 



dx~^' dy ~ ' dz~^' , (C) 



dBz dly _ dlx dcz _ dcy dlx __ 



dy dz ' dz dx ~ ' dx dy ~ 



Integrating this system of equations, which may be effected without diffi- 

 culty, we find, 



Lv = a + Bz - Cy, 



hy — b + Cx — Az, 

 is = C + Ay - Bx ; 



the well-known expression for the displacements of a rigid body. These 

 being the most general values which hx, cy, Zz admit of, the truth of the first 

 theorem is evident. 



With regard to the second theorem, if the body is so constituted that the 

 displacements cx, ly, cz must satisfy the equations 



'^ - 4 ^-« dhz_^ 



dx ~ ' dy ~ ' dz ~ ^' (D') 



^+^^2A' —+^ = 25' '^+ — ^2C' 

 dy dz " ' dz dx ' dx dy 



A, B, C, a, b, c, being functions of a;, y, z, the body will have at each point an 

 infinite number of inextensible directions situated on the cone (real or ima- 

 ginary), 

 -4 cos^a -f 5cos^/3 -hCcos^Y -t- 2^'cos j3 cos 7 -t- 2B'cos 7 cos a -f 2 C'cosa cos^=U. 



If the constitution of the body be given. A, B, C, &c., will be given func- 

 tions. In this case the equations (D') furnish the means of determining 



3 B 2 



