COVARIANTS OF PLANE CURVES. 
1181 
6 , Deduction of the conditions in the form of linear 'partial differential equations. 
For this purpose, it will be convenient to consider the independent parts of the 
conditions separately, and the headings of the paragraphs indicate the portions 
considered. 
only. 
Multiplying factor = 1 — (d d^ w )B,Y, 
Therefore 
77 — p becomes 77 — P — B^Y^ (tt — P); 
^ — X becomes ^ — X + B^ {tt — P + Y^^ (^ — X)]. 
of 
= S2.C,.iY^Y, 
+ {:r-P + Y,(f-X)ig^^j 
-J + 1 ^ ~ + <^2 + OU/ • 
Since Yj does not occur explicitly in the function, we must equate separately the 
parts which contain Y^ and those which do not contain it. The symbol S will now 
be used instead of % to denote that Y^ does not occur within the summation. 
Thus 
a/ 
a/ 
~ B) 0 _ p) “ ~ 0 (^1 _ X) ~ — S (n + 1) Y„ 
aY„ 
( 11 ). 
Since f is homogeneous of degree of in tt — P and the differential coefficients, 
we may write the first of these equations 
= . ( 12 ), 
which expresses analytically a condition already found. 
There is no other condition necessary in order that a function may be a covariant 
for the ordinary Cartesian transformation, which leaves the relations of a curve with 
infinity unaltered, that is for a change of coordinates. 
To proceed, we have 
