210 MATHEMATICS: G. M. GREEN 
class of systems of partial differential equations which are a direct 
generalization of the single ordinary homogenous linear differential equa- 
tion of the nXh order. A system of the kind referred to contains a single 
dependent variable and any number of independent variables, and has a 
fundamental system of solutions, that is a definite number of linearly 
independent solutions in terms of which any other solutions of the 
system of differential equations is expressible Hnearly, with constant 
coefficients. 
The following discussion is concerned throughout with functions of 
p independent variables. If any of the variables be complex, we shall 
suppose the functions to be analytic in those variables. However, we 
shall state all theorems for the case in which the independent variables 
are real, and the functions either real or complex; the modifications 
which must be made if some or all of the variables be complex are easily 
suppHed, and will need no further mention. We shall impose upon the 
functions no restriction other than the existence of certain partial deriva- 
tives in a certain connected ^-dimensional region A of the space of the 
independent variables. 
Let yu jiy . . . , % be functions of the p independent variables 
Wi, W2, . . Up. We shall denote by y'i \ yf \ etc., partial derivatives 
of y%, of any kind or order whatever. It will be unnecessary to specify 
just what derivative of yi is denoted by y^p. However, in any given 
discussion the same superscript (J) will denote the same derivative 
throughout. If a derivative y^'^ exists for each one of the set of functions 
yi (i = Ij 2, . . .jn),we shall say that the set of functions possesses 
that derivative. We may now state the fundamental theorem concerning 
the linear dependence of functions of several variables: 
Theorem I. Let the set of n functions yi, yi, . . yn of the p 
independent variables Wi, W2, . . Up possess enough partial derivatives , 
of any orders whatever, to form a matrix. 
' y yn 
• y Jn 
• 7 yn 
of n-1 rows and n columns, in which at least one of the (n~i) -rowed 
determinants, say 
.fi) 
y?' , # , 
(„-2, (»-2, 
