INDETERMINATE EQUATIONS 
469 
1. Double equation of the first degree. 
The equations are 
a x + a — u 2 , 
/3x + b = w 2 . 
Diophantus lias one general method taking slightly different 
forms according to the nature of the coefficients. 
(a) First method of solutioji. 
This depends upon the identity 
12 f JL 
5 12 
ìip-qW = M- 
iiip + q) 
If the difference between the two expressions in x can be 
separated into two factors p, q, the expressions themselves 
are equated to {^\p + q)} 2 and {p — q)] 2 respectively. As 
Diophantus himself says in II. 11, we ‘ equate either the square 
of half the difference of the two factors to the lesser of the 
expressions, or the square of half the sum to the greater 
We will consider the general case and investigate to what 
particular classes of cases the method is applicable from 
Diophantus’s point of view, remembering that the final quad- 
raticfin x must always reduce to a single equation. 
Subtracting, we have (a — /3)x+ (a — b) = u 2 — w 2 . 
Separate (a — (3)x + {a — b) into the factors 
P, {{oi—fi)x + {a—b)}/p. 
We write accordingly 
u + w = («^±(a-h) 
P 
u + VÜ = p. 
u + w — 
Thus u 2 = ocx + a = 
therefore {(a — fi)x + a — b+p 2 ] 2 = 4 q) 2 {o(x + a). 
This reduces to 
{(x~f3) 2 x 2 + 2x{{oc—(3) [a — b}— p 2 (a + /3)} 
+ [a — b) 2 — 2 p 2 (a + b) + p 4 = 0.