1862.] 15 



may be so prepared as to lead to this equation directly. To effect 



this, it suffices to eliminate from one of these equations - , from the 



dx n 



other - -, and to reduce in each the coefficient of the one which 

 dx n -\ 



remains to unity, and then apply the theorem (2). 



3. Between (I.) and (III.) and between (II.) and (III.) the same 

 process may be applied as between (I.) and (II.). The effect 

 of this is to give new partial differential equations ; in fact, to 

 generate a system which will be complete when the further applica- 

 tion of the method gives rise to no new equations, but only to 

 identities, or to repetitions, or combinations of the equations already 

 obtained. And though any equation of the system may be combined 

 with any other, according to the theorem, in order to form a new 

 one, yet it may be shown that the system will be complete when no 

 new equation arises from the combination of any with the original 

 ones (I.), (II.). 



4. Suppose that in this way m partial differential equations 

 have been obtained, including those two into which the given 

 system of ordinary differential equtions was transformed. Then 

 that system of ordinary differential equations will admit of exactly 

 nm integrals, i. e. the number of integrals will be equal to the 

 number of the variables diminished by the number of partial 

 differential equations. 



5. To determine these integrals, let the complete system of partial 

 differential equations be represented by 



A 1 P=0,A 2 P=0, ..A WI P=0; 



then multiplying the second by X a , the third by X 3 , &c., and adding, 

 we have 



a single partial differential equation, which, X 2 X 3 . . \ m being regarded 

 as indeterminate, will be equivalent to the system of equations from 

 which it is formed. Represent this equation by 



then its auxiliary system of ordinary differential equations will be 

 dx^__dx^ __dx n 



