1921-22.] Concomitants of Quadratic Differential Forms. 155 
where e is so small that its square and higher powers are neglected, and 
where the four functions tj, 0 are perfectly general and entirely 
independent of one another. It follows that the infinitesimal transforma- 
tions can be taken also in the form 
V = Xg, X^), 
= X^ — €^(Xj ^ , a?2 5 ^’3 5 ^ 4 ) ) 
^35 ^ 4 )- 
For the infinitesimal variations of the coefficients of the form and of 
their derivatives, as well as for the derivatives themselves, it is convenient 
to define here the symbols that will be used. In the case of any function 
■\jr of the variables , x ^ , x ^^ , whether ^jr be tj, 0 or any of the co- 
efficients a, . . ., 71 , we shall write 
/ _ r, I 
11. The infinitesimal variations of the variables x.^, x.^, x^, x^ are given by 
dx-^ = (1 — — €$2 dx^ — dx^ , 
dx^ = — dx^ + ( 1 — €7]2)dx2 — cTjg dx^ — erj^ dx^ , 
dx^ = — eJi dx^ - 6^2 dx2 + (1 — e^^dx^ — €^4 dx^ , 
dx^ = — dx^ — g^2 dx2 - ed^ dx^ + (1 ~ ^O^dx^ . 
Hence the modulus of transformation, H, is 
1 - , 
- els . - 
^^4 
1 - ^’?2 . 
- 
-e?2. 
1-^fs, - 
-e^i, 
-602- 
- €03 , 1 - 
-^4 
— 1 — €(|^j + ’’?2 + ^3 + ^ 4 ) ’ 
on neglecting squares and higher powers of e. 
We at once have the transformations of the variables X^, Xg, Xg, X^ in 
the form 
(X,',X/, X3', X/) = ( 
-e^2, 
-^^4 : 
^ x„ X2, X3, 
X4). 
-erji, 
- = 
-€ 7^4 
-e ^4 
-€0„ 
-e^ 3 . 
1 - 
The similar transformations of the var 
tables U4 , 
,U2,Ug^ 
, U4 are 
(U/, U2',U,', U/) = (l+€^,, 
€0, $ U„ U2, U3. 
V); 
1 + e 7 ? 2 , 
€02 
f +^£3? 
6^3 
1 + €^4 
