91 



If in any point invariant 'b, the values of x^, jj ; x,, y , . . . . , taken from 

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

 in the ascending powers of dt2, dtj, dt,, . . . dt„, the successive coefficients 

 of the separate powers of dt^, dtg, ..., and of tlie products dt2, dt^. ... 

 are all invariant functions of x', x", x'", ..., (y'), (y'^j, ij'"''), ...• Tliese 

 separate invariant functions may then be changed by means of equations 



dv „ _ d^y 

 (3) above so tliat only x', x'^, x''^, x'^, .... and y' = 3—' ^ — ^-^» » 



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 <p had been derived. 



8. The Differential Inrarianfs for the General Projectire Group. 

 The general projective group: p, q, xq, xp — yq, yp, xp + yq, x^p + 

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



Q= 



Substituting in Q the series expansions of .Xj, y2. X3, yg, ... x^, y^ 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} =^ an + aj , I - J -j- ^2 1^ _■''- J + a.3 , -i J 4- • ■ . . 



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

 degree /, and where 



11 = X' y" — x" (y' r:r y" x'^, 



12 = X' {y'^O — x''' (yO = y''' (x 



+ 3y'' (x'j^x^^ 



.-rj-, I.j — x' (y'^) — x*^ (y^) - y" (x^s j_ 6y''^(x')3x'' + 3y^^ x'(x^O* 

 I^ = x^' (y"'0 — x''' (y'O = 7''' (xO^x^' — y^' (x')'x''' -f By'^x' ^x' 



and so on until all orders of dilTerentials y', y", y'^', . . . . , y^"' have been 

 included. Now the separate ratios I2 : Ii, I3 : Ii. It : Ii. ■ • • ■ , axe separate- 

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

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



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



