87 



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

 we have* 



xi = 4.(x.y,ao) + [^'^Z^).5a+...., 

 •-- d ao J 



yx = t(x,y,ao) + [^^iZ^«->lc5a+ 



L d ao J 



But "l- (x, y, ao) = x, t (s, y, ao) = y, hence 



xi = X + I _— j '5 a + , 



^d ao J 



L d ao J 



c^x= {^t]As,^ ... =f (X, y) r5t+ ...., 

 i.d ao ' 



ryy = f iLt 1 ,u 4- • • • = V (X, y) f' t + 



Id ao J 



Omitting iufiuitesimals of higher order we have the rehitions 



d X = f (X, y) '5 t, <S J = II (X, y) (5 t 



as the infinitesimal transformations of a one-parameter grou]). 



In the notation of Lie the symbol 



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

 receive the increments (5 x, (5 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 



U /■ = s^x, y) p + '/ (X, y ) q. 



The infinitesinal transformations of an 7-parameter group would be given 

 by the symbol 



Uk / = fk (X, y ) p + vk (X, y) q, k z= 1, 2, 3, ... /•. 



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

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

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

 Jacobi's bracket 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 

 >- d X J 

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



