OF TESTIMONIES OE JUDGMENTS. G47 



The general proof consists in showing, that if the proposition is true for a par- 

 ticular value of n, it is true for the next greater. Whence, being true for the 

 case of n=l, 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 »=l. 



Let n=2, then we have to consider the system (2), which may be reduced to 



the form 



axy + bx 

 1 ,=p (3) 



axy + bx + cy + d v ' 



axy + cy ... 



,=q (4) 



axy+bx + cy + d 



Let us represent by Y the variable value of the first member of (4), when x and 

 y are supposed to vary in subjection to the single condition (3). We have then 



axy + bx + cy + d ^ ' 



Now differentiating (3) and (5) relatively to x and y, we find, after slight re- 

 ductions, 



(ay + b)(cy + d)dx+(ad-bc)xdy = .... (6) 



" dYJ ad ~^dx ^ ax + %f X + d) dy ... (7) 



where, as before, Y=axy + bx + cy + d. Substituting in (7) the value of dx found 

 from (6), we have 



, v _ ( ax + c ) (&# + d) { a V + &) ( c y + d)~ (ad— bc) 2 xy , 

 (ay + b)(cy + d)Y 2 J 



The numerator of this expression may be reduced to the form 



V (abcxy + abdx + acdy + bed) 

 whence 



dY _abcxy + abdx + acdy + bed 



dy~ (ay + b)(cy + d)Y ^ ' 



m 



This represents the differential coefficient of Y taken with respect to y as indepen- 

 dent variable, x being regarded as a function of y determined by (3). The ex- 

 pression is always positive, if x and y are positive. 



Now let y vary from to oc through the whole range of positive magnitude. 

 Writing (3) in the form 



Ax , ftx 



where A=ay + b, B=ey + d, the quantity #must, by reference to the case of »-=l, 

 have a positive value, since A and B are positive and p fractional. Whence, as 



dY 

 y varies from to oc , the value of -j- is always positive. 



Now when y=0, Y=0, and when y= a, Y=l, as is evident from (5). Therefore, 

 as y increases from to ex , Y continuously increases from to 1. In this variation 



