424 



Mr. R. F. Gwyther. On the Differential [May 4, 



Or, briefly, we may say the result is the interchange of the operators 

 Q x f?nd 2 . Regarded as functions of A 2 , A 3 , &c, the coefficients in a 

 contravariant are developed from a matrix satisfying Q 2 F = 0, by the 

 process which has been already described. Regarded as a function 

 of a 2 , &3, &c., they are derived from a matrix satisfying Q x f = 0, by a 

 process which differs from that previously described by the inter- 

 change of the operators Oi and Q 2 . Any function of the coefficients 

 in a covariant function which would be an invariant for a change of 

 coordinates, but not for a homographic transformation, will be the 

 matrix of a contravariant function. The matrix of the reciprocal is 

 the discriminant of the highest group of homogeneous terms in n 

 and f treated as a binary qu antic. 



§4. 



To apply these results to the cubic osculating the standard curve 

 at the point y), remove the origin temporarily to that point. The 

 condition of intersecting the standard curve in a number of points 

 coincident with this temporary origin is an invariantal relation, and 

 we may treat the coefficients in the equation as if they contained 

 differential invariants only. 



The form of the matrix of the cubic is 



f< + (0 + X L 6 + 2 L 6 2 ) Ui + yr + + f 2 L 6 * + f 3 L 6 3 + ^ JV, 

 with the condition /+% 8 02 + tt e*V r 4 = °> 



where all the functions contain differential invariants only, and, 

 retaining only invariantal coefficients, the expanded equation is 



yJs7r z — U. i 2 U 5 f 1 7r 2 g+ (0 + U 5 2 f 2 ) 7r^ — U 2 6 U 5 (0x + u 5 2 f 3 ) P 



—ui^Tt 1 + u^u^tt^— u 2 7 (2/+ ^ 5 2 02) £ 2 + Kfr — o, 



and writing the differential invariants of the several orders u 2 , u 5 , 

 U 7 , U 8 , U 9 , we have for the shape of the standard curve near the origin 



U 7 g7 tT 8 + 10%% 8 % 2 U 9 -3U 7 2 

 u 2 u 2 u 5 ufu^ U 2 b U 6 A 



Substituting in the equation of the cubic, we find 



fa = ^3 = Y^ = = /» = — % 2 01> / = ~ *V02, 



and % 2 = U 7 /, U,^ = Y 8 f, 



where V 8 stands for U 8 + 8% 4 . 



If the cubic is non-singular this determines all the functions in 

 the matrix, and gives the differential equation 



