614 PROFESSOR BOOLE ON THE COMBINATION 



or, for simplicity, in the form 



U=0. 



The lower limit of u is, by (11) and (12), either or^ — ^ , according as 



the latter quantity is positive or negative ; the upper limit of u is the least of the 



?? -4- {7 ~J- 7* ~— " X 



quantities p, q, r; suppose it p. First, let - — ^ De positive, then, making 



u equal to this quantity, the value of U, as given in the first member of (14) 

 becomes negative. Again, let u—p, then U becomes positive. Thus, as u varies 



from Pjl3_JHzz_ U p i p f \j changes from negative to positive. Now 



— — =(2u— p — q — r + 1) 2 + 4:u(2u— p — q —r + l)+(p — u)(q — u) + (q — u)(r — u)+ (r—u) 



Ct/VV 



(p-u) . (15) 



which within the supposed limits is always positive. Hence U varies by continu- 

 ous increase, and once only in its variation becomes equal to 0. 



<Y\ -L. /y _1_ rp "1 



Secondly, let *- — ^ be negative, then u, varying from up to p, U as 



before will vary by continuous increase from a negative to a positive value. See 

 the first member of (14). Whence U, changing by continuous increase from a 

 negative to a positive value, will still only once become equal to 0. 



Wherefore, in either case, one root only of (10) will lie within the limits as- 

 signed to u in (11) and (12). And this one value substituted in (9) will give one 

 set of values for s\ t\ v. 



20. The solutions which we have now obtained of the same problem on dif- 

 ferent hypotheses with respect to the selection of the simple events, set in clear 

 light the principles upon which the due selection of such hypotheses depends. 

 The hypothesis which seems most readily to present itself utterly fails, while 

 the other, based quite as much upon an apparently remote speculation on lan- 

 guage, as upon the study of the laws of expectation as usually conceived, finds a 

 support and confirmation within the realm of pure mathematics which is of the 

 most remarkable kind. 



21. A practical simplification of the general method is suggested by that step 



of the preceding solution, which reduces (5) to the form (7). If we remove the 



traces ( ' ) from the letters in the latter system (and they do not at all affect the 



solution), we obtain what (5) would become if we replaced each of the symbols 



s, t, v, by unity. Practically, therefore, we may modify the general rule in the 



following manner : — Having obtained V, replace each of the symbols 7, t, &c, by 



unity, and proceed with the reduced value of V just as before, i .e., let V s 



represent that portion of V of which s is a factor, &c, then form the system of 



equations 



Yi = r . . . = A ±£ C =V .... (J) 



p q u 



