OF TESTIMONIES OR JUDGMENTS. 649 



Axy + Bx 



V 



Aocy + Cy 



= P 



= q 



v 



which corresponds with the form of the general system (3) (4) in the case of «=2. 

 Whence, for each positive value of z, one positive set of values of x and y will be 

 found. The system (11), (12), (13) admits, therefore, of one solution in positive 

 val lies of x, y, z, and of only one. 



To prove the proposition generally, it ought to be shown that the function 

 exemplified in the first members of (10) and (14), for the cases of n~2 and n=3 

 possesses universally the same property of consisting only of positive terms. I 

 have proved that it does for the case of n=4, and the analysis was such as to leave 

 no doubt whatever of its general truth. 



I will now offer a few remarks on the application of the above proposition. 

 The system of equations for determining s and t . ., Art, 21, is of the form 



-i = — . . = V . . . . (15) 



p q 

 V being a function of the same general character as the one discussed in the 



foregoing proposition, but with this difference, that its coefficients, if we regard 



it as a complete function, are all equal either to 1 or to 0. 



Thus in Art. 18, we have 



V=stv + s + t + v + l 



Here the terms st, tv, and vs, must be considered as present, but with the coeffi- 

 cient 0. 



This limitation does not affect the essentially positive character of the deter- 

 mining function exemplified in (10) and (14). Whence the system (15) cannot 

 have more than one solution in positive values of s, t, &c. This shows that the 

 solution of the system of equations furnished by the general method can never be am- 

 biguous. 



The vanishing of some of the coefficients of V does, however, affect the rea- 

 soning by which it has been shown, that for the general form of V discussed in 

 the last proposition, one solution of the algebraic system in positive values will 

 exist. Thus Y in (5) does not vanish with y, if both b and d vanish. And gene- 

 rally this vanishing of coefficients in V entails conditions among the quantities 

 p, q, r . ., in addition to that of their being fractional, in order that the derived 

 algebraic system may admit of a solution in positive values. 



Thus if we take, as in (7) Art. 18, 



V = 8tV + S + t + V + l 



with the derived algebraic system 



stv + s stv + 1 stv + V 



-y- =p ~v~ =<7 ~^r~= r 



VOL. XXI. PART. IV. 8 M 



