[ 437 ] 
XXX. On Simultaneous Differential Equations of the First Order in which the Number of 
the Variables exceeds by more than one the Number of the Equations. By Geoege 
Boole, F.B.S., Professor of Mathematics in Queen's College, Cork. 
Received June 19, — Read June 19, 1862. 
It is a fundamental proposition of analysis that a system of n ditferential equations of 
the first order containing n+1 variables admits of n integrals, each of which is 
expressed by a function of the variables equated to an arbitrary constant. 
But when a system of n difierential equations of the first order connects n-\-r vari- 
ables, r being greater than unity, no existing theory assigns in a general manner the 
number of theoretically possible integrals of the above species, or shows us how to dis- 
cover them. Yet such cases are of great importance. 
I wish to develope here the theory of a method for the solution of the above classes 
of equations, which was published by me in the ‘ Proceedings of the Eoyal Society ’ for 
March 6th of the present year, and which enables us to assign the number of theoreti- 
cally possible integrals, and to reduce their discovery to the solution of a system of 
simultaneous difierential equations equal in number to the number of integrals, and 
expressible as exact difierential equations. 
The solution of the problem as thus reduced may be efiected by known methods, but 
I have thought it desirable to discuss this part of the subject also in direct sequence to 
the other, and in conformity with its method. 
Of the Connexion between ordinary and qmrtial Differential Equations. 
It has been found convenient, in researches bearing upon the general theory of dif- 
ferential equations, to use the term ‘ integral ’ in two distinct senses, viz. to denote, as 
above, a relation satisfying the difierential equation or system of equations, and expressed 
by the equating of a function of the variables to a constant, and to denote the function 
itself. The particular sense intended will always be shown by the connexion. 
With this convention two systems of difierential equations will be said to be equi- 
valent when they have in either of the above senses (and the one implies the other) 
the same system of integrals. This will explain the meaning of the following pro- 
position. 
Peoposition I. — A system of n ordinary differential equations of the first order con- 
necting w+p variables may be converted into an equivalent system of r linear qmrtial 
differential equations of the first order. 
Let 0 ^ 2 , .. . x„+r be the variables, then the supposed given system of differential eqna- 
MDCCCLXII. 3 0 
