INDETERMINATE EQUATIONS 
473 
, , x 2 + ax + a = u 2 ) 
(2) 0 9 2 1 
par + a = w j 
(The case where the absolute terms are in the ratio of a square 
to a square reduces to this.) 
In all examples of these cases the usual method of solution 
applies. 
ax 2 + ax — u 2 1 
/3x 2 +bx — w 2 \ 
The usual method does not here serve, and a special artifice 
is required, 
Diophantus assumes u 2 = m 2 x 2 . 
Then x = a/(m 2 — a) and, by substitution in the second 
equation, we have 
/3 (—^^ —? which must be made a square, 
\mr — a/ m 2 — a 
or a 2 /3 + ba(m 2 — a) must be a square. 
We have therefore to solve the equation 
abm 2 + a (aft — ab) = y 2 , 
which can or cannot be solved by Diophantus’s methods 
according to the nature of the coefficients. Thus it can be 
solved if {afi — ab)/a is a square, or if a/b is a square. 
Examples in VI. 12, 14. 
(b) Indeterminate equations of a degree higher than the 
second. 
(1) Single equations. 
There are two classes, namely those in which expressions 
in x have to be made squares or cubes respectively. The 
general form is therefore 
Ax n + Bx n ~^ + ... + Kx — L — y 2 or y z . 
In Diophantus n does not exceed 6, and in the second class 
of cases, where the expression has to be made a cube, n does 
not generally exceed 3,