304 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



the application of the rule, no two minor limits of distinct terms 

 can be added together, for either those terms will involve some 

 common constituent, in which case it is clear that we cannot add 

 their minor limits together, or the minor limits of the two will 

 not be both positive, in which case the addition would be useless. 



PROPOSITION IV. 



9. Given the respective numbers of individuals comprised in 

 any classes, s, t, *c. logically defined, to deduce a system of nume- 

 rical limits of any other class w, also logically defined. 



As this is the most general problem which it is meant to dis- 

 cuss in the present chapter, the previous inquiries being merely 

 introductory to it, and the succeeding ones occupied with its ap- 

 plication, it is desirable to state clearly its nature and design. 



When the classes s 9 t..w are said to be logically defined, it 

 is meant that they are classes so defined as to enable us to write 

 down their symbolical expressions, whether the classes in ques- 

 tion be simple or compound. By the general method of this 

 treatise, the symbol w can then be determined directly as a deve- 

 loped function of the symbols s, t, &c. in the form 



1 

 w = A + + - C+-D t (1) 



wherein A,B,C, and D are formed of the constituents of s, t, &c. 

 How from such an expression the numerical limits of w may in 

 the most general manner be determined, will be considered here- 

 after. At present we merely purpose to show how far this object 

 can be accomplished on the principles developed in the previous 

 propositions; such an inquiry being sufficient for the purposes of 

 this work. For simplicity, I shall found my argument upon the 

 particular development, 



-rf + o(i-o + J(i-*)* + J(i-) (i-O, (2) 



in which all the varieties of coefficients present themselves. 



Of the constituent (I -s) (I - <), which has for its coeffi- 

 cient -, it is implied that some, none, or all of the class denoted 



