1921-22.] Concomitants of Quadratic Differential Forms. 207 
43. From the first of this equivalent set, we have 
so that 
and therefore 
2S8,/ + 2^ft'8,' + 2iy,'a/ = 0 
P 7 
+ + y ’ 
. , ni 
where m is a constant, and /m is independent of a?/. But while and /3 
are functions of x^' only, y is a function of x^' and x ^ ; hence 
— = function of only = ^, 
and therefore 
-rr|. 
y=Vf.) 
where r is a function of X-^ only and /x a function of x^ only. 
With this value of y, we have 
7l ^ 7ll 5 7‘22^M22^j 
and then the second equation of the equivalent set becomes 
/^22 J_4_^ El _1_^ El-L Ey ^ = 0 
/X /3^~^ f3 T 13 T S 
Now V, S are functions of x^' only, while ju is a function of X 2 only ; 
hence, in order that this equation may be satisfied, we must have 
^ = constant = , 
and therefore 
/X = H sin (?/X2 + K) 
where H and K are constants. Manifestly, H can be absorbed into r ; and 
so we now have 
while 
y = r sin (nx2 + K) , 
&Ei; . ^i^+EE 
/3T~^J3 S TS (3'^ 
The third equation of the equivalent set becomes 
r f3S 
and the fourth equation of the equivalent set becomes 
r/ 8/ 
