INDETERMINATE PROBlfeMS, 
105 
tion for m, is positive or negative. But, besides this, there 
is another limit, or 'particular value of q to be determined, 
which is of great use in finding the number of answers. 
It is evident from the given equations, that the 
'Values of x will begin to be negative, when z is so 
increased as to exceed $ T~V-£; and that those of 
m 
y will, in like manner, become negative, when z is 
taken greater than : therefore, as long as 
continues greater than ^ (supposing the 
value of q to be varied) so long will x admit of a 
greater assumption for 2 than y will admit of, without 
producing negative values; and vice versa. By making, 
therefore, these two expressions equal to each other, 
ng—ml 
ng — ml 
the value of a will be given (=r 
am -f nb 
c 
expressing the circumstance wherein both the values of 
<v and ?/, by increasing z, become negative together. 
But this holds only when m is a positive quantity; 
for, in the other case, the last term (— mz) in the ge 
neral value of x being positive, the particular values do 
not become negative by increasing, but by diminishing 
the value of z ; it being evident, that no such can re 
sult from any assumption for z, but when q is greater 
than -y-» 
o 
To apply these observations to the equation, 7x -f- 
9y + 23z — 9999, proposed, we shall, in the first. 
whence the least value of y is given = 5; and the 
greatest of x — 1422. Again, from the equation am -f 
bn — c, or 7m + 9/? rr 23, we have m ~ 3 — n — 
o/i — 2 . . .. 
—— ; m which the least positive value of n is given 
~ 1 ; and the corresponding value of m — 2; and so 
the. general values of x and y do here become 1422 
O 2