426 Mr Bateman, Solution of a system of differential equations 

 The solutions are then 



(Xo - Xi) (X, - Xi) (Xi - Xo) (X3 - X,) 



+ 





Xo (Xi - Xo) (X, - X3) 



o X2 XgWp _^ ^ 



"(x,-xO(x,-x,)(x,-Xiy 



(Xj — Xg) (X3 — X2) (X4 — Xg) 



Xi Xo yin , . 



-I 1__2_0 ^_},^f 



(Xj — X3) (X2 — X3) (X4 — X3) 



_! XiXgXaTlQ ^_^^^^ 



(Xi — X4) (Xa — X4) (Xg — X4) 



The solution for the case of w— 1 products is given by 

 N = 'Ecre-^'-*, 

 where the constants c,. are obtained by expressing 



X1X2X3 ... \ n—i 



CC (x A- Xi) (x + X2) ... (X + \n) 



in partial fractions. 



The method by which the solution of the system of differen- 

 tial equations has been obtained is really of very wide application 

 and may be employed to solve problems depending on a partial 

 differential equation of the form 



dt \dx' dy' dz' '") ' 

 provided the initial value of V is known. 

 For if we put 



u{s)=\ e-'*V{t)dt, 

 Jo 



f^ dV 

 su{s)-Vo= e-''irdt, 

 Jo ot 



it appears that u{s) satisfies the partial differential equation 



^(8l'3i8i-)^+»+''« = « W- 



Further, if V satisfies some linear boundary condition which 

 is independent of t the function ti will generally satisfy the same 

 boundary condition. This function (u) must be obtained from the 



