Motion of Solid and Fluid Bodies. 163 
Hence, equations (15) are equivalent to 
(a—1)be + (B — m)b/c! + (c — nbc" 
+ (a + Bi + 1)E + (a, +B, + 12)! + (as + Bs + 15)0" = 9, 
(a —L)ac + (B — M)a’c’ + (c — n)a”’c” 
+ (a+ Bin )m + (a+ B+ Y2)m! + (a;+ Bry, )m” = 0, (16) 
(a — L)ab + (B — m)a’b’ + (c — n)a’b” 
+ (4 + Bi 1) + (4 + Be +2)0 + (4; + Bs + 15)2” = 0. 
It is possible to find a unique system of rectangular coordinates for which these 
equations shall be true; for assume the ellipsoid, whose equation is 
(a—1L)a@ + (B—n)y’* + (c —N)2? 
+ 2(a4,+ 8+ )yz + 2(a, + B+ Y2)22 + 2(a,+ 6, +7;)ry = 1, 
and transform it by the formula, 
«=a +’ +27, 
yaar + by’ + eZ, 
ze aaa’ t+ by'+ eZ, 
the coefficients of the rectangles will become the expressions (16), and, conse- 
quently, for axes of coordinates coinciding in direction with the axes of this 
ellipsoid, the equations (15), (16) will be satisfied. 
The function y,, referred to these axes of coordinates, will have the number 
of its constants diminished by three, and consequently the total number will ulti- 
mately be twelve ; it does not appear possible to banish a greater number by any 
transformation with rectangular coordinates, as in this transformation we have 
only three independent variables ; with oblique coordinates, however, we should 
have six, and it is quite possible that there should exist in solid bodies an oblique 
system of axes, which would simplify still further the form of the function v,. 
Such a system of axes, as well as the rectangular system just determined, would 
have an intimate connexion with the molecular structure of the body; though it 
is not easy to see exactly what that connexion would denote. 
It may be useful to notice here a remarkable relation which exists between 
the function v, and a certain surface of the fourth order, whose properties have a 
great similarity to those of the function; and the connexion between them is 
