DIFFERENTIAL EQUATIONS OF THE SECOND OEDER. 53 



upon the subsequent determination of the arbitrary functions in the integral equiva- 

 lent of the system. A question therefore arises as to how an integral equivalent 

 can be obtained. At first sight it seems that, as the number of equations (being 

 five) is equal to the number of unknowns (I, m, n, ?>, ?) to be determined, HAMBURGER'S 

 method* might be applied to our special instance, though not when the Dumber of 

 independent variables in the original equation is more than three. But, as a matter 

 of f;ict, one of the equations of the system is a functional consequence of two others ; 

 viz., the equation 



dm dn dl dn 



dx dy "~ dy " dx 



is a functional consequence of 



dv dv 



d~ V = l + n ?> Jy = m+ ^ 



It thus follows that there are only four equations independent of one another 

 involving the five variables ; consequently HAMBURGER'S method does not apply. On 

 the other hand, the inference is that, as the equations are fewer in number by unity 

 than the number of variables to be determined, one arbitrary element must exist 

 in any general integral equivalent. This arbitrary element and other arbitrary 

 functional forms, by the foregoing theory, are determined by means of the equation 



dv dz 



- = n - > 



nil. du 



so far as they can be made determinate. 



It is therefore necessary to seek for some integral combination of the subsidiary 

 system, apparently without at present having any perfectly general process of 

 constructing such a solution. It may, however, be pointed out that, as there are 

 four independent equations involving five quantities, they can be used to determine 

 four of them in terms of the remaining one or, more symmetrically when this is 

 possible, to express all five of them in terms of some variable. When such expres- 

 sions have been obtained, they are to be substituted in 



dv dz 



T~ = n T ' 

 du du 



the full solution of the resulting form of which equation will then serve to determine 

 the quantities. 



We proceed to consider one or two examples in connection with the foregoing 

 theory and explanations, dealing particularly with well-known equations. 



* CEELLE, t. Ixxxi. (1876), pp. 243-281 ; ib., t. xciii. (1882), pp. 188-214, the number of indepen- 

 dent variables being two, and the number of equations being equal to the number of dependent 

 variables. 



