1115 

 put (/ z= const, and integrate according to Jacobi's method. If 



/ii = 0,, H^=C^, iln — Cn (5) 



is a set of normal integrals of the equations, from which />, may 

 be solved, the characteristic function 



V=^i:Fidqi, (6) 



may be formed, where the functions F{q,, c,, a) represent the quan- 

 tities /?, deduced from equations (5), and putting 



T- = t, A~ = ^» ....— = r„ (7) 



OC^ Of, OCn 



these will be the additional integrals, where 



x,=t-}-c* t, = c* .... r„ = c* (8) 



The quantities Ct* are the 7i integration-constants. The iso-parametric 

 problem is thereby solved. 



3. The differential equations of the rheo-parametric problem. In 

 order to obtain these equations we shall pass from the variable 

 quantities pi and qi to the variables Ci and ti . This is a "contact- 

 transformation". It is obtained by means of the characteristic function 



V{q 

 as transformation-function 



uCi,a)= \ i:Fidqi (6') 



dF bV 



Oqi OCi 



The differential equations retain the Hamiltonian form. If rz remains 

 constant, the new Hamiltonian function is equal to the transformed 

 old one, i.e. to c< and the following trivial result is obtained: 



Ci = 0, c, = 0, c. = ; ti = 1, t, = tn = 



We now ajlow a to change, i.e. we put a = function {t). The 

 transformation-function V is now an implicit function of t through 

 the intermediary of q^ , c, a7id a : 



dV fdV- dV-\ dV- 



dt i \Oqi OCi ) oa 



The differential equations (3) retain their form all the time, but 

 the new Hamiltonian function K now becomes 



ir=. +r^V (10) 



