MATHEMATICS: G. M. GREEN 
211 
IF, ^ 
yi 
y2 
y'^ 
yr'\ 
yn-i 
yl'li 
vanishes nowhere in A. Suppose, further, that all of the first deriva- 
tives of each of the elements of the above matrix M exist, and adjoin to 
the matrix M such of these derivatives as do not already appear in M , to 
form the new matrix 
M' = 
, y2 , 
yi 1 y2 i 
(?) 
, yn 
> yn 
which has n columns and at least n rows, so that q ^ n — 1. Then if 
all the n-rowed determinants of the matrix M' in which the determinant 
W„ is a first minor vanish identically in A, the functions yi, y2, . • 
y„ are linearly dependent in A, and in fact 
yn = Ciyi-{-C2y2-\- 
+ Cft-iyn-U 
the c^s being constants. 
The proof of this theorem is very similar to the familiar one of Fro- 
benius for functions of a single variable [Cf. M. Bocher, Trans. Amer. 
Math. Soc, 2, 139-149 (1901)]. For p = l, the theorem becomes a gen- 
erahzation of the ordinary Wronskian theorem for functions of a single 
variable, and includes the latter theorem as a special case. 
It should be noted that for fimctions of several variables it is not 
possible to define a single determinant which may properly be called a 
Wronskian; however, a Wronskian may be defined for a completely 
integrable system of partial differential equations, of the kind mentioned 
above. Before giving this definition it will be convenient to state an 
existence theorem for the system of partial differential equations. 
Let us call a set of derivatives y, yO-\ yC^^ . . . , y^""^) of a fimction 
y a normal set, if for every element y(^^ of the set there exists in the 
set at least one other element y(«), whose order is one less than the order 
of y(r)j and from which yi^) may be obtained by a single differentiation. 
The existence theorem referred to may be stated as follows: 
