646 PKOFESSOK, BOOLE ON THE COMBINATION 



tuating my own views upon important questions connected with this subject are, 

 that I should regret having engaged in inquiries so lengthened and laborious 

 as those of which I now take leave, if I did not think that as materials for future 

 judgment, they may possess value and importance. And although the interest 

 attaching at present to these inquiries is chiefly speculative, it may be that they 

 will yet be found to possess a practical utility. The vast collections of modern 

 statistics seem to demand some kind of reduction. I am sure that all who read 

 this paper will feel that even towards this end I regard the labours of the mathe- 

 matician as contributing only in a secondary degree. 



APPENDIX A. 



The following proposition in Algebra is of extreme importance in connection 

 with the theory of probabilities. It was originally published by me in the Phi- 

 losophical Magazine for March 1855 ; but the present paper would be incomplete 

 without some notice of it. 



Proposition. 



If V be a rational and integral function of n variables x, y, z . ., involving no 

 power of these variables higher than the first, and having all its coefficients posi- 

 tive, and being complete in all its terms, then if Y x represent that part of V which 

 contains x, Y y that part which contains y, and so on ; the system of equations 



I*=l2.. =V (1) 



P 2 



p, q, &c. being positive fractions, admits of one solution, and of only one solution, 

 in positive values of x, y, z . . 



To exemplify this proposition, let us suppose 



V = axy + bx + cy + d 



a, b, c, and d being all greater than ; then it is affirmed that the system of 



equations 



axy + bx axy + cy 



—^ = J q J = axy+bx + cy + d . (2) 



p and q being positive, admits of one, and only one solution, in positive values 



of x and y. 



The proposition is true when n=l. For then Y=ax + b and the system (1) is 



reduced to the single equation 



ax 



— = ax + b 

 P 

 Whence we have 



bp 

 as— 



a (1— p) 



and this value is positive if a and b are positive, and p a positive fraction. 



