170 BELL SYSTEM TECHNICAL JOURNAL 



nates, the initial components of velocity, and the final instant. It is 

 necessary now to consider W as depending upon the following equivalent set 

 of eight variables: the initial and final instants h and /2, the coordinates 

 (-V'n, .i"2i, A"3i) of the initial point, and the coordinates {xn, X22, x^-i) of the final 

 point. Regarding W in this manner, we at once obtain the following 

 relations from equation (24) 



dW „ dW 



-—- = -//2, T— = 7r„2, (25) 



012 OXn2 



^~ = Hi, -— = -TTnl, (26) 



where H2 denotes H[xi(t2), ■ • ■ , Triit^), • • • , ^2] and Hi denotes H[xi{ti), • • • , 



7ri(^i), • • • ) ^i]- 



Let us now consider the partial differential equation 



°*|^ + H{xi ,X2,x^, dW/dxi dW/dx2 , dW/dxz , /) = 0. (27) 

 at 



The preceding work shows that the function W we have been considering 

 (with .Yn, .T21, .V31, /i regarded as parameters, and with the symbols .V12, .T22, 

 .T32, h replaced by :Vi, .T2, .V3, / respectively) is a particular solution of this 

 equation. We shall show that the complete solution of this equation 

 possesses remarkable properties in connection with dynamical problems. 



The complete solution of equation (27) is a function of .vi, .V2, x-^, t, and 

 of four arbitrary constants, of which one is merely additive, and can be 

 neglected for our purposes. Let the solution be written 



W = IT(.Ti, .T2, .T3, /, ai, ao, as), 



where the a's are the three essential arbitrar>^ constants. 



We write the equations 



f = A, (28) 



can 



where the /3's are further arbitrary constants. These equations implicitly 

 determine the x's as functions of / and the six arbitrary constants ai, • ■ ■ , |5s. 

 We also write the equations 



f = ... (29) 



dXn 



These equations determine three functions x„ of the .v's, the a's, and /. 

 In virtue of equations (28), the x's are ultimately functions of /, the a's, and 

 the jS's. 



