1893.] 



Co variants of Plane Curves, 



421 



Simple Conditions. 



Call the algebraic degree of the equation in f rj, d, and the algeb- 

 raic degree of a coefficient in a z , a 2 , a 3 , . . . . , d x , which we shall call the 

 degree of the coefficient. In the same coefficient call the sum of the 

 indices of differentiation the weight of the coefficient, and write it w. 

 Write D for the determinant of the homographic substitution, 



A, = (AX + BY + l) (1 + BxYO-CX + BxY) (A-fBY x ), 

 H = AX + BY-f 1, 

 v = Ag+B v +l. 



Then (1) the form of the multiplying factor is 

 D d+d * . \- . . v ~ d ; 



(2) £, 17, x, y, and a x only enter in the forms x and y—y — 



ai (£—#)> which will be written w— p ; 



(3) The function is homogeneous in rj — y and the several differential 



coefficients of y ; 



(4) The weight of the coefficient of each power of £— x is uniform, 



i.e., each coefficient is isobaric, and this weight, diminished 

 by the index of the power of x, is uniform throughout 

 the function ; 



Conditions in the form of Linear Partial Differen tial Equations. 



(5) The other equations of condition take the forms 



8/" 3f 



a ' , = SSjpay^-H ir- (A) 



d(£— x) * - da n 



^-x){a-x)^ f —+^- r )^ f —-df} = SO^K-xJ^- 

 L 0(f— x) 0{7r— <p) J ca n 



(B), 



= SS (p — l) a p a n - P (C), 



Oa n 



where S denotes summation for all values of n which introduce 

 values of a n from a 2 upwards, and SS denotes a similar double 

 summation, first with regard to which introduces values of a p 

 between a 2 and a n , and secondly with regard to n, which introduces 

 values of a n from a 2 upwards. 



