1921-22.] Concomitants of Quadratic Differential Forms. 209 
and therefore 
(This equation for S emerges from a similar treatment of the expression for 
r and substitution in the second equation.) 
We now liave two equations for S alone, viz. the equation just obtained 
and the former equation, which can be taken in the form 
= :i ) + 
JL V. 
Im 8^* ’ 
and they must be consistent with one another. Now the constant I occurs 
only in the latter equation, which is of the second order ; hence the other 
equation, which is of the third order, must be consistent with 
2/,q 
NK 
' 8 '2 
i°n 
_ v_. 
dx -^ l\ 
4 - 
88/ 
with 
1 
rH 
1 — 1 
8n' 
4'2 ] -1- 
IT ^111 
381,'“ 
8/3 
88/ 
^) + 2 
8/3 
5 8/3 
88/ 
^ V _f_ 2/^-2 V _ Q 
In this equation, substitute the former value of viz. 
-^111 _ 
8/ W8/ 
S.' 
after reduction, we find 
- 2/tY' _ ^/.r-hr = 0 
But we had the relation 
88/2 
82 
,V! 
83 
1^]] ^ _ 7.2 ^ 
^8/2 ^ §2 S 
1 A 
Im 82‘ 
consequently we must have 
Lm 
We cannot have m infinite, for then both /3 and T would be infinite; we 
cannot have I infinite, for then /3 would be infinite and T would be zero ; 
and we exclude the case 8^' = 0, which now is trivial (and it would require 
that m = 0). Hence we must have 
h = 0 . 
The analysis can be reversed, step by step ; therefore, the two equations 
for 8 are consistent with one another, if this condition is satisfied. There 
then is only a single equation for 8 ; it is 
I and m being constants. 
VOL. XLII. 
_ 4 ^ 1 . _ 4?L = 0 
8/2 ^ 8 
14 
