DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 11 



a relation that is satisfied identically in virtue of the defining equations. As it 

 is deduced from two of these equations, and it is satisfied identically in virtue of 

 others, the inference is that the set of equations must consequently be reduced 

 by one -in number when only those which are independent are to be retained. 

 Treating the other three in the same manner, we find 



dk 'dff da dff 



db df dh df 



do; " dx dy " dy f ' 



df dc dff dc 



dx dj~ dy dy J 



equivalent to the identities obtained in 3. As these are deduced from the eight 

 defining equations, they are satisfied in virtue of those eight ; they do not constitute 

 any addition to the aggregate. Seven, therefore, is the number in this class. 

 The remainder are the equations characteristic of F = 0, viz., they are 



A=0, f=0, r, = 0, =0, 



being four in all. It thus appears that the tale of independent partial differential 

 equations in the system is eleven, being the same as the number of quantities to be 

 determined. 



It is to be noted the verification is simple that the original equation 



F = 



is an integral of these eleven simultaneous equations. Hence, for their effective 

 solution, other ten integrals would be required if further considerations cannot be 

 introduced ; but it will appear from examples that this can be done, having the effect 

 of appreciably shortening the process of integration. 



6. One generalisation is immediately suggested by the results obtained. In solving 

 equations of the second order in two independent variables, the subsidiary system is 

 composed of a set of simultaneous equations involving one independent variable in 

 effect ; and the preceding investigation shows that, for equations of the second order 

 in three independent variables, a subsidiary system can be constructed in the form of 

 a set of simultaneous equations involving two independent variables in effect. It is 

 thus suggested and it is easy to see that the suggestion can be established definitely 

 that, for an equation of the second order involving n independent variables, a subsi- 

 diary system can be constructed in the form of a set of simultaneous partial differential 

 equations of the first order involving in effect n 1 independent variables and a 

 number of dependent variables, this system being subsidiary to the integration of the 

 proposed equation. 



c 2 



