18 NATURE AND DESIGN OF THIS WORK. [CHAP. I. 



than to say, that, like the previous method, it is based upon the 

 employment of the " final logical equation," and that it definitely 

 assigns, 1st, the conditions which must be fulfilled among the 

 numerical elements of the data, in order that the problem may 

 be real, i. e. derived from a possible experience ; 2ndly, the nu- 

 merical limits, within which the probability sought must have 

 been confined, if, instead of being determined by theory, it had 

 been deduced directly by observation from the same system of 

 phenomena from which the data were derived. It is clear that 

 these limits will be actual limits of the probability sought. 

 Now, on supposing the data subject to the conditions above as- 

 signed to them, it appears in every instance which I have exa- 

 mined that there exists one root, and only one root, of the final 

 algebraic equation which is subject to the required limitations. 

 Every source of ambiguity is thus removed. It would even seem 

 that new truths relating to the theory of algebraic equations 

 are thus incidentally brought to light. It is remarkable that 

 the special element of quantity, to which the previous discussion 

 relates, depends only upon the data, and not at all upon the 

 qucBsitum of the problem proposed. Hence the solution of each 

 particular problem unties the knot of difficulty for a system of 

 problems, viz., for that system of problems which is marked by 

 the possession of common data, independently of the nature of 

 their qucesita. This circumstance is important whenever from a 

 particular system of data it is required to deduce a series of con- 

 nected conclusions. And it further gives to the solutions of 

 particular problems that character of relationship, derived from 

 their dependence upon a central and fundamental unity, which 

 not unfrequently marks the application of general methods. 



17. But though the above considerations, with others of a 

 like nature, justify the assertion that the method of this treatise, 

 for the solution of questions in the theory of Probabilities, is a 

 general method, it does not thence follow that we are relieved in 

 all cases from the necessity of recourse to hypothetical grounds. 

 It has been observed that a solution may consist entirely of terms 

 affected by arbitrary constant coefficients, may, in fact, be 

 wholly indefinite. The application of the method of this work to 

 some of the most important qiiestions within its range would 



