91 



If in any point invariant </>, the values of x.,, j^; x.^, j^ , taken from 



(I) be substituted, and then the result developed into in infinite power-series 

 in the ascending? powers of dtj, dtg, dt^^, . . . dtu, the successive coefficients 

 of the separate powers of dtj, dtj, ..., and of tlie products dtj, dt^, ... 

 are all iavariaiit functions of x', x", x'^', ..., (j^), (y"), (y'"j, .... These 

 separate invariant functions may tlien be changed by means of equations 



cVy ^// _ d2y 



(3) above so that only x', x^', x' 



and y' = 



dx 



dx2' 



occur. Then by algebraic manipulation the parameters x', x'', x'^', 

 may be eliminated, leaving a differential invariant for the continuous 

 group from which the point invariant tp had been derived. 



8. Tlie Differential Invariants for the General Projective Group. 

 The general projective group: p, q, xq, xp — yq, yp, xp -\- yq, x^p -|- 

 xyq, xyp -j- y'-q, when extended leaves invariant the point-function. 



q: 



Substituting in Q the series expansions of X2, j^, x^, j^, ... Xj, j^ from 

 equations (I), and developing the determinants, we have the ratio of infinite 

 series which may be further developed into a single power series of the form 



Q^=a„ + aj, [^] + a, [^-^-j + a^ (^] + .... 



where ai is an expression containing a function of dtj, dtg, dt^, dtj to 

 degree /, and where 



(K) 



Ij := x' ly'^ — x'^ (y'. = y x'3, 



12 = x' (y"0 — x''^ (y^) = y''' (x')^ + Sy'' (x'l^x'^ 



13 = x' (y'M — x*^ (y^) - yi^ (x')^ + 6y'"(x^)3x'' + 3y'^ x'(x^O^ 



+ 4 y (x') ■'x"\ 



I^ = x'' [Y") — x'" {y") = y (xO^x^^— y [x')'^x"' -t 3y'^x' (x'' 



and so on until all orders of differentials y', y" , j"' , , y"" have been 



included. Now the separate ratios 1^ : Ii, la : lu I4 : Ii. r are separate- 

 ly invariant, and when reduced as in equations (K) contain the arbitary 

 parameters x', x'', x"' , x^"'. The elimination of these parameters is 



"See Pro. Ind. Acad., 1898, p. 135. 



