1893.] 



Covariants of Plane Curves, Sfc. 



423 



From this it is clear that d0 o , the coefficient of the highest power 

 of (ir— p), satisfies the differential equation Q 2 / = 0, and that the whole 

 expansion can be found if d0 o is known. 



On this account I call <*0 O the source or matrix of the covariant, 

 and investigate the solutions of Q 2 / = 0. 



It is next proved that 



*£-|<V= <»•-*>/» 



and hence that, if / is a homogeneous, isobaric function of a 2 , «3,. . . . , 

 such that Q 2 / = 0, and that 2w—d x = 0, then df/dx is equally a solu- 

 tion of Q 2 / = 0. 



If we know two solutions of Q 2 /= 0, solutions of all higher orders 

 can then be educed by the process here implied. 

 Q 2 = o on expansion becomes 



r\ f) f) 



a 2 ~+2a 3 ~+3a i ^ -\ =0, 



d«3 da>i oa 5 



and therefore we have two solutions readily, viz., 



u 2 = a 2 , 



u i — a 2 a i — a 3 2 ' 



The educts successively found by this method are, however, not 

 generally irreducible, and it is shown how to find the irreducible 

 solutions for the successive orders, which are written u 2 , u±, u 5 , &c. 

 Any common solution of Q 2 f = and Q x f = is a differential inva- 

 riant, and not the matrix of a covariant. For the second and fifth 

 orders, the seventh and all orders higher than the seventh, there is a 

 differential invariant, and for the sixth there is a common solution of 

 Q 2 / = and Q 3 f = 0, which I write L 6 . It is the matrix of a straight 

 line through x, y, and all matrices may be expressed as functions of 

 Ui, L 6 , and differential invariants. The order of the covariant can 

 be inferred from the mode in which u± and L 6 enter the matrix. 



§3. 



If the systems of coefficients are subjected to a reciprocal trans- 

 formation, of which tt— 7 = may be taken as the typical 

 relation, there is this relation between the operators Q x and Q 2 — if u 

 is a solution of either Q x f = or Q 2 / = in which a 2 , a 3 , . , . . , are 

 the arguments, and if, in consequence of the substitution for these 

 qualities of A 2 , A 3 , .... (similar functions of the correlative system 

 of coordinates), u becomes IT, then U is a solution of the other 

 corresponding equation, that is, of Q^W = or Q X F = respectively. 



