OF TESTIMONIES OR JUDGMENTS. 647 
The general proof consists in showing, that if the proposition is true for a par- 
ticular value of 7, it is true for the next greater. Whence, being true for the 
case of n=1, it is true universally. I will exemplify the method by showing how 
the truth of the proposition, when n=2 is dependent upon its truth when n=1. 
Let n=2, then we have to consider the system (2), which may be reduced to 
the form 
any + bu 1 
axy+batey+d ? ; : : z ; ; @) 
any + cy bs 
axy+bateyta 2 E ’ 5 ; ; ; (4) 
Let usrepresent by Y the variable value of the first member of (4), when z and 
y are supposed to vary in subjection to the single condition (3). We have then 
any + cy 
“any +ba+cy +d (5) 
Now differentiating (3) and (5) relatively to z and y, we find, after slight re- 
ductions, 
(ay +b) (cy +d) dx+(ad—be) ady=0 ‘ i : : (6) 
d—b w+e)(be+d 
avy-@ v3 Wan + ty Gyan : : (7) 
where, as before, V=ary+be2+cy+d. Substituting in (7) the value of dz found 
from (6), we have 
__(aa+e) (ba +d) (ay+b) (ey+d)—(ad—be)ay 
ae (ay +b) (cy+d) V? dy 
The numerator of this expression may be reduced to the form 
V (abcay + abda + acdy + bed) 
whence 
dY _abexy+abda + acdy + bed 
dy (ay +b) (cyt d)V 
(8) 
This represents the differential coefficient of Y taken with respect to y as indepen- 
dent variable, z being regarded as a function of y determined by (8). The ex- 
pression is always positive, if z and y are positive. 
Now let y vary from 0 to « through the whole range of positive magnitude. 
Writing (3) in the form 
Aw 
Au+B? 
where A=ay+b, B=cy+d, the quantity z must, by reference to the case of n=1, 
have a positive value, since A and B are positive and p fractional. Whence, as 
(9) 
y varies from 0 to «, the value of = is always positive. 
Now when y=0, Y=0, and when y= «, Y=1, asis evident from (5). Therefore, 
as y increases from 0 to «, Y continuously increases from 0to 1. In this variation 
