OF TESTIMONIES OR JUDGMENTS. 613 



The condition to which s' t' v are subject obviously is, that they shall be posi- 

 tive quantities, for this is equivalent to the condition that s, t, v shall be positive 

 fractions. 



From (7) we readily find 



s' _ J!_ _ _j/_ _ s't'v' 

 p — u q — u r — u u 



Whence 



= s't'v' + s' + f + v' + l . (8) 



, p — u 



s = 



2u— p — q — r + 1 



2u—p — q — r+ 1 

 r — u 



v = 



(9) 



2m— p — q — r + 1 i 



Substitute these values in the equation 



s' s'ifv' 



p — u u 



and reducing we get 



(p — u) (q — u) (r — u) = u (2m— p — q — r + 1) 2 . . (10) 



an equation for determining u. 



And now let us inquire into the consequences which flow from the condition 

 that s' t' v are positive quantities. 



In the first place, the last member of (8), and therefore each other member 

 of that system will be positive. This requires that the denominators, p—ti,q-u, 

 r—u, and u, should be positive, whence we have 



= tt >° = ) .... (11) 



u<p, u<q, M<r J v. t 



Again, p—u, q—u, and r—u being positive, the common denominators, 2u-p-q— r 



+ 1 in (9) must be positive, whence 



= p + q + r— l /10A 



«>^ .... (12) 



Such are the conditions relative to u. They agree in all respects with those as- 

 signed in the previous investigation, in (5), Art. 13 ; and, as in that article, the 

 elimination of u leads to the conditions of possible experience, 



p>0 q>0 r>0 \ 



P > q + r ~\ .... (13) 



q> p+r—1 



r> p + q— 1 > 



It may be well to notice, that these conditions involve the necessity of p, q, and r 

 being fractional, though of course this does not exhaust their significance. 



19. It remains to show that when the above conditions are satisfied, the 

 system (7) will admit of but one solution in positive values of s, t', v', and that 

 (10) will furnish but one value of u satisfying the conditions (11) and (12). 

 Let us write 10 in the form 



u(2u— p — q— r+1) 2 — (p — u) (q — u) (r— u) = . (14) 

 VOL. XXI. PART IV. 8 C 



