310 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



formed by selection or aggregation. Hence, for the minor limits 

 of z we have the values c l p l and c 2 p z . 



13. It is to be observed, that the method developed above 

 does not always assign the narrowest limits which it is possible 

 to determine. But it in all cases, I believe, sufficiently limits the 

 solutions of questions in the theory of probabilities. 



The problem of the determination of the narrowest limits of 

 numerical extension of a class is, however, always reducible to a 

 purely algebraical form.* Thus, resuming the equations 



let the highest inferior numerical limit of w be represented by 

 the formula an (s) + bn (#)..+ dn (1), wherein a, 6, c, . . d are 

 numerical constants to be determined, and s, t, &c., the logical 

 symbols of which A, B, C, D are constituents. Then 



an (s) + bn (t) . . + dn (1) = minor limit of A subject 



to the condition D = 0. 

 Hence if we develop the function 



as + bt . . + d, 



reject from the result all constituents which are found in Z), the 

 coefficients of those constituents which remain, and are found 

 also in A, ought not individually to exceed unity in value, and 

 the coefficients of those constituents which remain, and which 

 are not found in A, should individually not exceed in value. 

 Hence we shall have a series of inequalities of the form f< 1, 

 and another series of the form g < 0, /"and g being linear func- 

 tions of , b, c, &c. Then those values ofa,b..d, which, while 

 satisfying the above conditions, give to the function 



an(s) + bn(t) . . + dn(l), 

 its highest value must be determined, and the highest value in 



* The author regrets the loss of a manuscript, written about four years ago, 

 in which this method, he believes, was developed at considerable length. His 

 recollection of the contents is almost entirely confined to the impression that the 

 principle of the method was the same as above described, and that its suffici- 

 ency was proved. The prior methods of this chapter are, it is almost needless 

 to say, easier, though certainly less general. 



