MATHEMATICS: G. M. GREEN 
213 
The Wronskian just defined has properties similar to those met with 
in the theory of an ordinary differential equation. Thus, it may be 
shown without difficulty that 
n-l 
« j=0 
W^4'-'\ (^=1,2,. . .,p) 
so that we may determine by a quadrature a function / such that 
|^=Va<>« {k = l,2,. . .,p) 
Therefore, the Wronskian W of n solutions of the completely integrable 
system may be determined by a quadrature from the coefficients of the 
system, and is given by the expression 
W= const. ^, 
This is a generalization of the theorem of Abel for an ordinary homo- 
geneous linear differential equation of the nth order. 
We shall state one more theorem, the analogue of a famiHar one 
concerning an ordinary differential equation. The completely inte- 
grable systems to which it applies are of somewhat less generality than 
those for which the existence theorem has been given. 
Theorem III. Suppose the completely integrable system considered 
in Theorem II has in addition the following properties: 
P. The set of primary derivatives is such that, if yO) be any one of 
the set, then all the derivatives of lower order from which yd^ may be ob- 
tained by diferentiation also belong to the set. 
2°. All the first derivatives of the primary derivatives exist for each of 
the np coefficients aj-^**^ (/ = 0, 1, . . . , n — l; k = \, 2, . . ., p). 
Then the system of differential equations may be transformed in but one 
way into a system 
n-l 
bUb 
^ai''*'y' (y = 0,l,. . .,»-l;fe = l,2,. . .,p) 
ik=l,2,. . .,p) 
for which all of the quantities 
- „_i 
are zero, by the transformation of the dependent variable y = \y, where 
X = const. 
