90 



invariants of a given transformation are known, then all others may be 

 found by diiferentiation.* 



An ^--parameter group Uk/ extended to include the increments of 

 y^, y", . . . y"', when equated to zero, gives *■ partial differential equations 

 in r -|- 2 variables. These r equations have two independent solutions, ^i 

 t^) y. y'. ■ • ■ y'""^). ^2 (^1 y> y'> • • • y*^'^')> which are differential invariants of 

 the ?--parameter group. After the plan here indicated Lie has calculated 

 the differential invariants for the twenty-seven groups of the plane. 



Tlie calculation of differential invariants may be made by an entirely 

 different method than that used by Lie, and indeed without any knowledge 

 of the group extended as indicated above. A knowledge of the form of a 

 point invariant for the group is necessary. 



Let a point invariant o iXj, j^, k^, J2, ■ ■ •) be given, and suppose the 

 points Xi, yi ; Xj, yj ; . . . ; Xn, yu, to be located upon a plane curve 



X = fi (t), y = ±"2 (tt. 

 Then we would have 



Xi =f, >ti), J, =f, itj), ... x„ = fi {t„\ y„ =f, (tn). 



Allowing Xj, y2 ; X3, yg ; . . . ; Xu, yii to coalesce toward xi vi, we may then 



expand x^, J2, • • • ■ in power-series 



/X2 = xi + x'dt2 + x^^dt| + ..., y2 = y, + (y^dt2 4-(y'^idt| + ..., 



(^M X3=xi + x'dt3+x^^dt|+..., y3=yi + iy0dt3-(y'')dtl+..., 

 ( 2 2 



and so on for x^, y j, . . . x,,, y„, where 



/ix / dx, ,, d^x, /,/ d^x, 



(1) x^ = — ', X ^ .-= 1, x^^ = i, , 



dtj dti^ dti^ 



(2) (yO=^, (y-)=^, (y"') = ^' 



dt 1 dtf dt^ 



The notation of (1), (2) should be changed from parameter notation to the 



ordinary y' ^ ^, y'^ = — ?1, 



dx dx2 



y'^— ^ = '■^ ' , hence (y') :^ j^ x', similarly, 

 dx x^ 



3 , (y^O=y''(x')'+y'x''; (y'^O =y''' (xO M- 3y''x' x'^ + y' x-^ 



'^ (yiv) = yiv (xO * + Qj''' (xO ^ x" + 'Sy'' (x'O ' + 4y" x' x'" + y' x'^ ; 



(y^) = yv (xO ' + lOyi^- (x') ^x" + y" (15 x' (x") = + 10 x'j ^x"') + 

 y^' (10 x'^ x'" + x' X" ) -|- y' x", 

 and so on for higher derivatives. 



■'Lie, Math. Annalen, Bd. XXXII. 



