PROFESSOR CAY LET’S TENTH MEMOIR ON QUANTICS. 
G21 
376. The fundamental property is : Every capital covariant, say I, lias for its leading 
coefficient the corresponding covariant i multiplied into a power of a : and this follows 
as an immediate consequence of the foregoing genesis of A. The co variant i of tlie 
form q (a, b, c, d, e, f X r E vY h as a leading coefficient =.y (a 3 cf— a~de + &c.) which, 
when a, b, c, cl, e, f . . . i denote leading coefficients, is —i into a power of a : and 
upon substituting for the quintic the linear transformation thereof 
(1, 0, c,f a 2 b — 3m, — ?/) 5 , 
(observing that in the transformation rj into — bp, aq the determinant of sub- 
stitution is =a), the value is still —i into a power of a ; or using the relation a — a, 
say the value is —i into a power of a. Now the covariant i is the same function of 
the covariants a, b, c, d, e, f that the leading coefficient i is of the leading coefficients 
a, b, c, cl, e,f; hence the italic letters now denoting co variants the leading coefficient 
still is = i into a power of a : which is the above-mentioned theorem. 
377. To show how the transformation is carried out, consider, for example, the 
covariant B : this is obtained from the corresponding covariant of (a, b, c, d, e, f'Xfh v) J 
that is 
vY- 
by changing the variables, and for the coefficients 
a, b, c, d, e, f 
writing 1, 0, c, f orb — oc 1 , a~e — 2qf ; 
thus the coefficients 
are 
First. 
Second. 
Third. 
l(a 2 6- 
- 3c' 2 ) 
\{a~e — 2 c/’) 
— ±c{a~b — 3c 2 ) 
+ 3c 3 
+ 2 cf 
+ 3/ 3 
= a% 
= cde 
= — 4a 2 6c + 1. 2 c 3 
+ 3 ( — (del -f- a 2 be — 
= ( i 2 ( — 3 ad — be) 
and we have thus the expression of B (see the Table No. 97); and similarly for the 
other capital covariants C, D . . . V, W : in every case the coefficients are obtained 
in the standard form ; that is, as rational and integral functions of a, b, c, d, e, f linear 
as regards f 
378. It will be observed that there is in each case a certain power of a which 
explicitly divides all the coefficients and is consequently written as an exterior factor : 
ae 
1 
af 
1 
bf 
I 
bd 
— 4 
be 
— 3 
ce 
— 4 
c 3 
+ 1 
cd 
+ 1 
d 3 
~f~3 
