124 Mr. R. F. Gwyther on 



If we put $ for U and <p for u, omitting the terms in 



v and w, we obtain the correponding terms for , /.t! +r $. 



dX?d i q d/i 



In the first case to be considered, where xp (x, y, z) is 



to be a scalar invariant, we have, writing the small letters, 



and putting ti p , q , r as an abbreviation for ^ ^ , 



d d d d 



dw 



d 



2— - y-r- + W-j- - V-j- + 



dy dz dv dio 



+ I 9. u p,q-l, r+1 — ru p,g+l, r-1 f " 



+ U V P, 2-1, r+1 - W ft 2 +l, r-1 + «r 2, Ad^~ r 



J \d 



+ 1 9 W p, 2-1, r+1 ~ TW p, 2+1, r-1 y p, 2, *■ X dw 

 j^A 2-1. r+1 " %, 2+1, -1/^77 



+ 



with two similar expressions, as the operators which act 

 as annihilators upon fa We will write these conditions 



0^ = \ 



Ot\p = I . (5). 



£l 3 xP = ) 



In the second case, when fa (x, y, z), fa (%,y, z)> fa { x ,y, z ) 

 are to be the components of a vector invariant, we have, 

 writing Q, h 12 2 > &3> f° r the same operators as before, 



&ifa = 0, ^1^2 = fa, ®afa = ~fa,\ 



n 2 \pi= -fa, a 2 fa= o, £i*fa= fa, I . . (6). 



£l 3 fa = fa } Q, s fa = - fa, £l 3 fa = 0. J 



The conditions sought are all expressed in these equations. 

 The expansion of these operators, even when we limit 

 ourselves to the second order of differential coefficients, is 

 somewhat long, but I will give the expansion for fti leaving 

 the others to be expanded by symmetry. Thus, altering 

 the notation, 



