1186 
MR. R. F. GWYTHER OX THE DIFFERENTIAL 
8. Relation between the operator and djdx. 
In the functions under consideration, x and y only appear in ^ — x, tt — p, and the 
differential coefficients. 
Hence 
where 
and 
d 
dx 
D. 
9 /dr X 9 I a 9 
D^. + 9.^, say, 
9 • V 9 
bljn 
• (17)- 
None of the operators O^, Og, Og, contain 9/9^%. Also we have to bear in mind that 
the functions do not contain y (except so far as it enters through tt — pb), and there¬ 
fore, in dealing with general terms, we must exclude y.^ when it makes its appearance 
as if a derivative of y^ 
The division of djdx into two parts makes the steps somewhat easier, and the final 
results are 
(Oi - til) £ - £ (Oi - ^i) = 2/2 + d.) ■ 
Gs - «i) I - £ (O 2 - a) = - ( 2 i 0 -d.) 'f ■ ■ ■ ( 18 ). 
(O3 - fla) £ - £ (O3 - n,) = (Oi - a,) + y, (0, - n,) 
Hence if be a covariant, satisfying the necessary conditions, and if f' stand 
for dfjdx, it follows that 
(Oi - fit)/' = — 2/3(w + c* + 4 )/' 
(O 2 — flo)/' = - (2ii) — d,)f 
(O 3 - fla)/' = 0 
(19). 
Now f' satisfies the condition as to weight and degree necessary for being a 
covariant, simply d and d,c remain unaltered, while w is increased by unity. 
Hence if the weight and degree of f are such that 
then f will also be a covariant. 
AYe are tlius led to the 
w -]- d -|“ dx — 0 
2w — dx = 0, 
