366 The Rev. J. H. Jellett on the Properties oflnexiensihle Surfaces. 



Ix, Zy, iz. Thus, for example, if the body be homogeneous, A,B, C, &c., will be 

 constants, and it is not difficult to prove that c.r, ly, iz will be of the form 



Ix = a.r + by + cz + d, 



hy = a'x + h'y + dz + d! , (E') 



Iz = a"x + h"y + c"z + d" . 



Let x\ y\ z', be the co-ordinates of the molecide in its new position. Then 

 since 



ix ^= X — X, 



^y = y' - y, 



iz = z' — z ; 

 we have 



x' = {a + 1) X + by + cz + d, 



y' = a'x +{b' +l)y + c'z +d\ (F) 



z' = a"x + b"y + (c" + 1 ) J + d". 



Hence it is easy to infer tlie following theorem : 



If a homogeneous body have at each point a cone of inextensible directions, and 

 if in the interior of the body there be described aii algebraic surface of any oi'dei\ 

 all the molecules situated upon that surface icill after displacement be situated 

 upon a surface of the same order. 



In general, whatever be the nature of the body, if ds be an element making 

 with the axes the angles a, j3, y, which satisfy the equation 



dhx , dly . , diz , 

 -r- cos- a + -pi cos^ B + — — COS^ 7 

 ax ay dz 



fdhy dc2\ , /dcz dix\ (dlx dly\ ^ ^ 



it is plain that we shall have 



Ids - 0. 

 Hence, 



Whatever be the laic of the displacement, there will be at each point of the body 

 an infinite number of directions {forming a cone of the second order), for ichich the 

 length of the element will be unchanged. 



We shall now return to tlie case of surfaces. 



