COVARIANTS OP PLANE CURVES. 
1187 
9. Theory of eduction of covariants. 
This theory is clear from what has preceded, but, to give it its simplest form, it is 
best to return to the preliminary conditions for a covariant. 
If is a covariant, such that iv f d dx = 0 and Tw — dx = 0 , the multiplying 
factor is v~‘^, and we may write the relation 
Then 
dX. d(f) 
dx dx d'K. 
Y)d+cL y 1 ^2 ^ 
d^ 
dX ■ 
Hence d^jdx is a covariant in which d and dx are unaltered and lo is increased by 
unity (3). 
Before making use of this theory of eduction it is necessary to find a method of 
finding some covariant functions from which the eduction must be made, and the 
method now to be explained will in general supersede this mode of eduction. 
10 . Method of development from a matrix. 
We have, however, another method for the formation of a covariant, namely, 
directly from the differential equations. 
Let d-m^ stand for the collection of terms algebraically homogeneous, and of degree 
d — m, and let d-m<f> 0 ’ d-mi>i, d-m^l^-z stand for the coefficients of terms containing 
— x)^, ^ — X, — xY among these homogeneous terms. Then, amongst these 
terms (15a) gives the law of derivation, namely, 
(7 + 4 * 7+1 — .(^^)’ 
a law which is independent of d — m, and for all values of d — m, 12 ^ d-yfpd-m — 0 . 
Hence, if we know ^4*0’ coefficient of the highest power of tt — p>, we find all 
the terms of the cZ* desfree. 
11 . Modes of derivation of homogeneous terms from those of a higher degree. 
The conditions (15b) and (15c) give us two modes of derivation, which imply the 
existence of certain relations and conditions. 
We have from (15b) and (15c) respectively, 
— x) + 2 + . . . 
+ + . . . 
(tt — p) + 2 d-T^ + • • • 
+ Hg + . . . 
+ ('^’^ + 1 ) d-m-f^ + • • •} 
+ d-m^ + . . .j = 0 . 
+ (m -f- 1) d-m-f^ + • • ■ ] 
+ d-m^ + . . . j = 0 . 
7 M 2 
( 21 ), 
(22). 
