the Imaginary Roots of Equations. 253 
cussion of Newton’s rules is that of Maclaurin, in the Philo- 
sophical Transactions, vol. xxxvi. No. 408, who has entered 
into a very long‘and elaborate investigation of them, the re- 
sults of which, however, only go the length of showing that 
‘* some imaginary roots exist in an equation,” whenever any 
of Newton’s criteria have place; and do not embrace the more 
general affirmation of the rules, that there are as many pairs 
of such roots as there are distinct criteria fulfilled. 
It is no doubt from the misgiving and uncertainty always 
attendant upon an undemonstrated principle, however nu- 
merous the individual instances in which it may have been 
safely trusted, that these rules of Newton have fallen into neg- 
lect, in the analysis of equations. It is one object of the pre- 
sent paper to revive and demonstrate them: another, and the 
more immediate one, is to prove the adequacy of the criteria 
already given to determine the true character of a pair of 
doubtful roots, as soon as by actual development we have 
reached the point where, if they are real, the separation of 
them must take place. 
It is desirable, when this stage of the approximation is ar- 
rived at, that we should be enabled to pronounce at once upon 
the nature of the roots under examination, from the conditions 
necessarily impressed upon the transformed coefficients thus 
attained, without having to apply to additional tests, or to exe- 
cute any new transformations or by-operations, for this pur- 
pose, as in the methods hitherto proposed. This object may 
be effected from the following considerations. 
I have elsewhere shown (Theory of Equations, p. 263), that 
when two roots, differing but little from equality, or concur- 
ring in several leading figures, are to be developed, these 
figures, after a certain early stage, will be furnished, one after 
another, by either of the two concurrent expressions which, in 
the arrangement below, stand vertically under the functions 
into which these roots first enter :— 
i GAN 'esricdpestenite 0)! cS, (alo: oc Fe) 
ap Rg AI oS,” 
n An See eeseeeerte 4A, 8 A; ZA, 
a 2A n—2 2A, AL 2 Ay 
(n—1) me wee eensese 3 As A, A, . 
and further, that when there is a discrepancy between the 
leading figures furnished by the two expressions used, the 
roots, if real, are about to separate. Now, without applying 
to any external source, or extending the development beyond 
