90 



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

 found by diflferentiation.* 



a<pi a.<pi 



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

 y^, y^^, . . . y<'', when equated to zero, gives r partial differential equations 

 in r -f 2 variables. These r equations have two independent solutions, (p^ 

 (X, y, y', ... y^'M. ^2 (^> y» 7^ • • • j'-'')) which are differential invariants of 

 tlie 7-parameter group. After the plan here indicated Lie has calculated 

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



The calculation of differential invariants may be made by an entirely 

 different metliod 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 ^ (x^, j^, x^, y2. ■• .) be given, and suppose the 

 points Xj, yi ; Xj, y^ ; . . . ; Xn, Jn, to be located ujton a plane curve 



X = f, (t), y = f2 (ti. 

 Then we would have 



Xi = fi ^tl), yi = f2 (ti), ... x„ = fj (t„\ y„ = f, (t,,), 



Allowing Xo, yj ; X3, y, ; . . . ; x„, y„ to coalesce toward xi yi, we may then 



expand x.^, j^, .... in power-series 



/ x,=:Xi + xMt. + x^^dt| + ..., y., = yi +(y'idt2 + (y'')dt|+..., 



(^M x3=:Xi + xMt3+x^^dt|+..., y3 = yi + (y')dt3- (y^')dt|+..., 

 ( 2 2 



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



Qx x^ _ dx, ^„ ^ d ^x, ^,,, _ d«Xi 



' ^ dt/ ■ dtj^' ' dti»' ■■■■' 



(2) (yO = ?^, (y'O = ^, (y'n - ^'^' 



dti dt^ dtf 



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



ordinary y' =:: ^Z, y z= _1ZL, .... 

 dx dx^ 



y^ = — ?^ — ^^ ' , hence (y') = y' x^, similarly, 

 dx X'' 



/ {r')=rur)'+rx''; (y^^O^-y^^'CxOM-Sy'^x-x-' + y^x-^ 



^ (yiv) = yiv (xO * + 6y (x') ' x'^ + 3y'' (x'O ^ + iy x' ^''' + j' x*^' ; 



(yv) = yv(x^) 5 -j- 10 yiv (x') ^x'' + y (I5x'(x'0 ' + iO.x') ^-x"') + 

 y'' (10 x'' x"' 4 5 x^ xiv) + y x\ 

 and so on for higher derivatives. 



'Lie, Math. Annalen, Bd. XXXII. 



