254 Prof. Young’s New Criteria for 
the stage thus reached, the character of the discrepancy ad- 
verted to will, of itself, decide the doubtful point; for the dis- 
agreement may consist either in the first expression giving a 
greater figure than the second, or the second a greater figure 
than the first: if the former happen, the roots sought will be 
real; if the latter, they will be ¢maginary. For it is plain that 
in the quadratic, to which the approximation tends (Equa- 
tions, p. 262), we shall have, for f(x), = 2; for, f, (7), m = 33 
for f, (w), n = 4, and so on; so that the conditions previously 
given supply, in these cases, the following criteria of the 
character of the roots sought, that is the roots are real or not, 
according as these conditions exist or fail:— 
Ay athens ib dele 
2A, A, 
A, A, 
Sige dg 
aie ge ke 
cA,” Bde 
Agei 2 An-s 
nA, (71) A,_y 
The value of these criteria, in connexion with the rapid 
mode of approximation taught in the work referred to, is ob- 
vious. In pursuing a pair of contiguous roots of f,, («) = 0, 
conformably to that method, we are to seek the development 
of the intervening root of /,,4; («) = 0, the successive figures 
of which, after a step or two, are always furnished by the con- 
curring expressions above, and are to carry on the work up 
to f (x); continuing the process as long as those expressions 
agree in giving the same leading figure. When this agree- 
ment ceases, the roots may be pronounced reai, if the first ex- 
pression eaceed the second; otherwise they will be maginary. 
And thus their character unfolds itself spontaneously, with- 
out any appeal to external tests or supplementary transforma- 
tions. 
The ultimate quadratic thus attained may with propriety 
be called the indicator of the doubtful roots; when it proves 
their reality, we may employ it to supply the leading figures 
of the two roots, which become distinct after the fnidtteate is 
reached. ‘This same indicator will also furnish an approxi- 
mation even to the remaining imaginary portion of the par- 
