157 
1921-22.] Concomitants of Quadratic Differential Forms. 
so we substitute for dx^, dx^, dx^ in terms of dx^, dx^, dx^, dx/ 
according to the law 
(dx^, dx^, dx^, dx^) = {l + ^ ^ dx^, dx^', dx^', dx^) , 
I €^1,1+ ^^2, 
I ^^15 ^C- 2 ’ l+^?3 5 ^^4 
! ^^1? ^^2’ ^^ 3 ’ l+^^4 
we equate the coefficients of the various power-combinations of dx^', dx^\ 
dx^, dxl, and, neglecting powers of e above the first, we find 
CL — (I — -+- - 1 - ” 1 " , 
h' - b = e(24^2 + + 2??i^2) j 
c - c = e(2p'^3 + 2/173 + 2c^3-f- 271^3), 
d' - d^ €(2/^4 + 21717 ]^ + H- 2 dd^ , 
/ - /==^(^^^2+.f'’?2 + ^?2 + ^^2) + ^(^^^3 + ^'^3+/^3 J 
9-9 = 3 + ^^'^3 + r/^3 + ^^3) + ^{9^1 + fvi + + ^^1) > 
hf — h = -f- biq^ +/^x ^^2 ^^^2 ^^2) ’ 
Z' - Z = e(a^4 -f- 74774 + ^7^4 + ^^4) + 
m - ?4i = + ^Vi +/^4 + ^^ 4 ) + + ^^^2 + ^^2 + ^^ 2 ) 5 
n' - n--^ €{g^^ +frj^ + c -h nd^) + e(Zf 3 + 771173 -f 77^3 -f 77^3) . 
13. Next, in connection with the differential invariants of the quadratic 
differential form and with the general body of its invariantive concomi- 
tants, we require the infinitesimal variations in the derivatives of the 
coefficients of the form with respect to x^, x^, x^, x^\ as already indicated, 
derivatives of the first order and the second order (but of no higher order) 
will be retained. These infinitesimal variations can be obtained as follows. 
Let any arbitrary small increments a^, a^, be made to x-^, x^, x^, x^ 
respectively ; and let the corresponding small increments of x-^, x^, x^, a;/ 
be a^, «2^ «4' respectively, under the general infinitesimal transforma- 
tions. We have 
X^ = X^-€^{X^, X^,X^, X4), 
+ otj' = + ttj - ef (Xj + ttj , ^2 + «'2 > ^3 + ®3 j ^4 + “4) } 
and therefore 
®i — ®i “ 5 • • • ) ~ ^(^1 5 • • • )}j 
so that 
®1 ~ ~ ^ ^2^2 “^3^3 ®'4^4 
+ 2(^11 5 ^22 ’ ^33 5 ^44 5 • • •? ^34 “'1 J ®2 ’ ^ 2 ’ ^4)“} ’ 
and similarly 
^2 - «2 = ~ + ^ 2^2 + + ^4'^4 
2(^11 ’ V22 ’ V33 ’ Vu 5 • • •? Vsi ^ 5 ®'2 5 “3 ’ “^4)^} i 
-^ 3 = - + ^ 2^2 + “sSs + 
^22 5 ^33) ^44 ^ • • -5 ^34^®^!) ^ 2 ’ ®3 ’ ^4)^} ’’ 
^4 ~ ^ 4 ~ ~ ^ 2^2 ^ 3^3 ®4^4 
"^2(^11’ ^22 5 ^335 ^445 • • *5 ^34^ 
^15 ^2 ’ ^3 ’ ^4)}' 
