CHAP. XX.] PROBLEMS ON CAUSES. 349 



here and in all similar instances, the function F, by the aid of 

 which the algebraic system of equations for the determination of 

 the values of # 15 . . x n9 15 . . t n is formed, is independent of the 

 nature of any function involved, not in the expression of the 

 data, but in that of the qutzsitum of the problem proposed. Thus 

 we have in the present example, 



Si (XT 1 ) 

 Prob. w = \= -, 



wherein V = 2! (XT) + S 2 (XT) + t } ..7 n 



= S (XT 7 ) +*!..*. (10) 



Here ^(XT) represents the sum of all symmetrical constituents 

 of the x and t symbols, except the constituent ~x\ . . ~x m ~f\ . . T n . 

 This value of V is the same as that virtually employed in the so- 

 lution of the preceding problem, and hence we may avail our- 

 selves of the results there obtained . 



If then, as in the solution referred to, we assume 



x\ ti x n t n #i e 



=-=r = m ly =-^r = wi n , =-=n l ,&c. y 



Xi t x X n t n Xi 



we shall obtain a result which may be thus written : 



M 



TT' 00 



M l being formed by rejecting from the function the constituent 

 Xi . . #, if it is there found, dividing the result by the same con- 

 stituent xi . . x m and then changing ^ into m ly ~ into m z , and 



#1 X 2 



so on. The values of M and N are the same as in the preceding 

 problem. Reverting to these and to the corresponding values of 

 m l9 m z , &c., we find 



Prob. w = M l (n + v - 1), 

 the general values of m r , n r being 



C r pr C r (l- Pr ) 



lll r , ll r - -r- -j 



fj.-t, r p r fJi-C r (l-pr) 



and fjL and v being given by the solution of the system of equa- 

 tions, 



