TO THE RESOLUTION OF PROBLEMS» 
105 
The numbers being denoted as above, we shall here 
have 7/—a| + 4- a 4 a] rr d, that is, 2a 4 4- 12ahr* 4- 
2a? 4 — d; from which, by transposition and division, 
x* 4- 6a 2 x z — id —a*; and, by completing the square, 
x 4 + 6a\v 2 4 9a 4 — jd 4- 8a 4 ; whence x 2 4- 3a 1 — 
s/\d 4 8a 4 ; and, consequently,xxz V-3a 2 4-s/\d 48 ah 
PROBLEM XLVIII. 
The surh, and the sum of the 5th powers of two num 
bers being given; to find the numbers. 
The notation in the preceding problems being still 
retained, we shall have 2a s 4- 20а 3 я а 4- 10a# 4 = e; and 
therefore a 4 4- 2 a 2 x- 
юа 
; and x 2 4- a z 
— i- ; whence 
10a 5 
* = VV 
e «4 a 4 
■ 4 
10a 5 
-a. 
'1 
nee 
PROBLEM XLIX. 
What two numbers are those, ichose product is 120 fa J, 
and if the greater be increased by 8 fbJ, arid the lesser 
by 5(c), the product of the two numbers thence arising 
shall be 300 (d) ? 
If the greater number be denoted by x, and the lesser 
by y we shall have 
xy — a, and 
a? 4- b x y-f c — d,by the conditions of the question. 
Subtract the first of these equations from the second, and 
you will have x 4 b x y 4- c — xy — d — a, that is, 
CT 4- by 4- be — d — a; where botfi sides being multi 
plied by x (in order to exterminate y), we thence have 
cx z 4 bxy -4 bex — dx — ax; but xy being — a,, there 
fore »; bxy — ab, and consequently, by substituting this 
value in the last equation, ex 2 4- ab 4- bex — dx — ax; 
whence c.r 1 f hex 4-ax — dx — — ab, and therefore 
x x 4- bx f — —; which, by making f — 
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