Permanent Forms of Mathematical Expressions. 1 23 



which give the position of the new axes, regarded as a rigid 

 system, relative to the original system. 



I shall, however, consider only a new set of axes 

 differing infinitesimally from the original axes, so that 



X = x + yd 3 — zd 2 , etc., 

 U = u + vd 3 — wd 2 , etc., 

 d d n d n d , , , 



^x = ^ + ^- ^' etc -' and$ = ^ 



I propose to consider the conditions under which a 

 function 



^(X, Y, Z, U, V, W,... $,...) or shortly i£(X, Y, Z) 



(whereby I wish to indicate a function of the co-ordinates, 

 the components of a vector function, and their differential 

 coefficients, and of a scalar function and its differential 

 coefficients) will be unaltered in form by the infinitesimal 

 change of co-ordinates : — and also the conditions under 

 which three functions of the same quantities (ip u ip 2 and ^ 3 ) 

 may have the character of three components of a vector 

 function. 



In the first case, \p(X., Y, Z) must become $(x,y, z) ; and 

 in the latter case, \pi(K, Y, Z) must become 



4>\( x , y, z ) + Q^{%, y, 2) - Mbfo y, 2), etc. 



The conditions are found immediately from considering 



dX p dY q dZ r dx p dy q dz 



c d p + q+r d p+qJrr \ 



+ 6l Y dx p dy q ~ l dz r+xU " r dx p df + W- lU ) 



I m d p+q+r d p+q+r d 



+ 02 \ r dx p+1 dfdz r ~ lU ~ P dx p - 1 dy"dz r+lU ~ dx p dy q dz* 



f dP+1+r dP+g+r d \ 



+ ° 3 X* dx p -Hy q ^dz" % ~ q dx p+1 dy q ~ 1 dz rU + dx p dy q dz rV )' 



