OF TESTIMONIES OR JUDGMENTS. 649 
Aay+ Be _ 
So 
Aay+Cy _ 
Taian ae ae 
which corresponds with the form of the general system (3) (4) in the case of n=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 
values 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 ¢. ., Art, 21, is of the form 
Vv. V 
= — oe _. = V 5 e = . 15 
S z (15) 
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 
Vestutett+utl 
Here the terms sé, ¢v, 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, ¢, &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 and d vanish. And gene- 
rally this vanishing of coefficients in V entails conditions among the quantities 
p.% 7 - ., im 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=stu+st+itu+1 
with the derived algebraic system 
stu+s __ stu+t_ Siurbun 
vor » ieee von 
VOL. XXI. PART. IV. 8M 
