87 



give to the point x, y an inliuitesinial motion. Expanding in powers of '5a, 

 we have* 



T / V , fd >1> (x, y, aoVi t . 



XI = -i- (X, y, a„) + \-!-^ '5 a + ..... 



•v d ao ' 



yx = t(x,y,ao) + !^^^-^^"M'5a+ 



L d a<, J 



But il' (X, y, ao) — x, t (x, y, ao) = y, hence 



>^d ao J 



yi = y ^ :, — '' a + ..... 



L d ao J 



rJx = f --•^- V? a 4- . . . = ^" (X, V) '5 t + . . . . . 



I d ao ' . 



<5 y = f' — '/' 1 '^ a + . . . = ?/ (X, T ) -S t + 



L d ao J 



Omitting infinitesimals of higher order we liave tho rohxtions 



rf X = f (X, y) rf t, (5 y — // (X, y) 'U 



as the infinitesimal transformations of a one-parameter group. 



In the notation of Lie the symbol 



denoting the variation which a function / (x, y) undergoes when x, y 

 receive the increments S x, S y, is employed as the symbol of an infini- 

 tesimal transformation. Writing p, q instead of the partial derivative of 

 ^' (x, y) with respect to x and y, respectively, we have 



!;/■= f (X, y) p + )/ (X, y) q. 



The infinitesinal transformations of an r-parameter group would be given 

 by the symbol 



Uk /^ ~ fk (X, y) p + ;/k (X, y) q, k ==: 1, 2, 3, ... r. 



4. TJie Group Criterion. One of Lie's fundamental theorems furnishes 

 a test whether or not any given set of infinitesimal transformations, Uk f, 

 k ^ 1 , 2, . . . r, actually forms a group. This test is the application of 

 Jacobi's braclrt expression 



Ui (Uj/)— Uj (Ui /■), (i, j == 1, 2, ... r, in all combinations). 



'■■"In this article the symbol will be used to denote the partial derivative of f with 



I- d X ' 

 regard to x, instead of the round d usually eiDployed. 



