CHAP. XX.] PROBLEMS ON CAUSES. 351 



wherein NI is formed by dividing by x v . . x nt and changing in 



the result =i into n ly ^ into n Z9 &c. 

 x l x z 



Now the final solution of the problem proposed will be given 

 by assigning their determined values to the terms of the fraction 



Prob. (x l , . . x n ) z Prob. w 

 Prob. (x 1 , . . #)' Prob. v ' 



Hence, therefore, by (11) and (13) we have 

 Prob. sought = g^. 



A very slight attention to the mode of formation of the func- 

 tions M l and N\ will show that the process may be greatly sim- 

 plified. We may, indeed, exhibit the solution of the general 

 problem in the form of a rule, as follows : 



Reject from the function $ (# 1} # 2 . . x n ) the constituent x l . . x n if 

 it is therein contained, suppress in all the remaining constituents 

 the factors x 19 x z , fyc. 9 and change generally in the result x r into 



CrJ)r . Call this result M,. 



fl - C r p r 



Again, replace in the function 0(#i, # 2 . .#) the constituent 

 Xi . . x n if it is therein found, by unity; suppress in all the remaining 

 constituents the factors r 15 x 29 *c., and change generally in the re- 



SU U 



V-C r (\~p r ) 



Then the solution required will be expressed by the formula 



fi and v being determined by the solution of the system of equations 



i v 1:= 0*-gij>i)'.0-gj>.) 



fl n - 1 



= {v-c 1 (l-p 1 ))..{y- Cn (l-p n )) 



y n-i 



It may be added, that the limits of /m and v are the same as in 

 the previous problem. This might be inferred from the general 

 principle of continuity; but conditions of limitation, which are 



