41] PARTIAL DIFFERENTIAL EQUATION. 135 



Equating the two values, 



by 38). 



Transposing and writing T + W = H, 



The function .0", the sum of the energies, depends upon the co- 

 ordinates q r and the momenta, p r = ^ If the force -function depends 



upon the time H will also contain t explicitly. Thus we have the 

 partial differential equation 



oo\ dS . 



+ 



The equation is of the first order since only first derivatives of S 

 appear, and, from the way in which T contains the momenta [equa- 



o ci 



tion 72)], is of the second degree in the derivatives Since S 



appears only through its derivatives an arbitrary constant may be 

 added to it. 



Thus we have the theorem due to Hamilton: If q lf . . . q m , ex- 

 pressed as integrals of the differential equations in terms of t and 

 2m arbitrary constants q^, . . . q&, p^, . . .p, are introduced into the 

 integral 94), and the result is expressed in terms of t, q lf . . . q m , 

 (Zi , . #m, then 8 is a solution of the partial differential equation 99). 



The converse of the proposition was proved by Jacobi, namely, 

 that if we take any solution of the equation 99) containing m arbi- 

 trary constants, q^, . . . q (other than the one which may always 

 be added), the equations 97) obtained by putting the derivatives 

 of S by the m arbitrary constants equal to other arbitrary constants, 

 p^j . . . pm w iU b e integrals of the differential equations of motion. 

 For the proof of this the reader is referred to Jacobi, Vorlesungen 

 tiber Dynamik, XX. 



Before giving examples of the utility of this method we shall 

 show that the arbitrary constants by which we differentiate need not 

 be the ^'s ? but may be any m constants appearing in the integral 

 equations. 



Suppose that in equations 91) we vary m of the arbitrary 

 constants c 1; . . . c m . We then have 



