1921-22.] Concomitants of Quadratic Differential Forms. 
161 
The Group of Characteristic Partial Differential Equations. 
16. According to the adopted definition of an in variant! ve function 
the relation 
(f>' ^cf) QP 
must be satisfied where (p' is the same function of the transformed quan- 
tities that occur in cp, as (p is of the untransformed quantities ; where O is 
the modulus of transformation ; and where p is an integer. Further, it is 
sufficient (after earlier statements) to take only the infinitesimal variations. 
For these variations, 
V.7q, .^2, .Tg, V 
= 1 “ df 1 + V 2 + ^3 + ^ 4 ) > 
and so the equation 
(/»(«/, . . ., n', Xf, . . ., x;, P/, . . ., p;, u/, . . ., Uf ) 
= {l-pe{i, + 7^, + ^, + d,)}cP(a, . . .,n, X^, . . ., X,, P,, . . ., P„ U^, . . ., U,) 
must be satisfied. And it is to be satisfied for all the infinitesimal varia- 
tions that can be imposed, however arbitrary^ arising out of the functions 
7], J, 6. It follows that, when we substitute the values of a', . . ., n\ 
Xj^', . . ., Xf, Pf, . . ., Pg’, Uf, . . ., Uf that have been obtained, and 
expand the left-hand side so as to retain only the first power of e, the 
coefficients of the various derivatives of r], 6 on the two sides must be 
equal. The terms, independent of e on the two sides, cancel one another ; 
the only derivatives, out of the total group of 136 arising from 7 ], 6 and 
occurring on the right-hand side, are , t]^, ^ 3 , 6^. Thus the coefficients 
of 132 of these, from the left-hand side, must vanish ; and there are four 
such coefficients arising out of the left-hand side, each of which is equal 
to —pf(p- 
The First Sub-Group: Eighty Equations. 
17. We consider the sub-groups in turn, dealing first with the sub- 
group of eighty that arise in connection with the third derivatives of 
7], 6. This sub-group is most conveniently managed by selecting the 
equations in the sets of four, that are furnished by taking those derivatives 
of f, 7], 6 which are exactly similar to one another. 
From the coefficients of we have 
2hJ-+ bj- + fj^ + 
VOL. XLII. 
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