determining the number of Imaginary Roots in an Equation. 115 



Or, expunging from each equation such numerical factors as 

 are common to all its terms, the equations will be 



n(n-l)(n-2) A ^ + 3(7i-l)(n-2)A l ^ 2 



+ 2 . 3(ra— 2)A 2 # + 2 . 3A 3 =0, 



(»-l)(n-2)(n-3) A 1 a? + 3{n-2)(n-3)2A^ 



+ 2 . 3(»~3)3A s a? + 2 . 3 . 4A 4 = 0, 



(n-2)(n-3)(n-4)2A 2 ir s + 3(»-3).(n-4)2 . 3A 3 # 2 



+ 2 . 3(n-4)3 . 4A 4 a> + 2.3.4. 5A 5 =0, 



&c. &c. &c. 



Now if any of these limiting cubics indicate imaginary roots 

 when tested by the criteria in the margin above, we may be sure 

 that imaginary roots exist also in the primitive equation. But 

 since only one pair of imaginary roots can enter a cubic equation, 

 it follows] that whether the criterion of imaginary roots be satis- 

 fied by the first three terms of any of the above cubics, or by the 

 last three, or by both sets of three, one pair of imaginary roots, 

 and one pair only, is necessarily implied. Upon examining the 

 coefficients of these cubics, we see that the first three terms of 

 each have a common factor ; so that in applying the criterion to 

 these three this common factor may be suppressed. Let the 

 common factor, namely (n — 3) in the second cubic, be sup- 

 pressed ; then the product of the first and third coefficients will 

 be 2.3 2 (ra — l)(w — 2)A,A a , and the square of the middle coeffi- 

 cient will be 2 2 . 3 2 (rc — 2) 2 A 2 2 . But these are the same results 

 as we should get by employing, in like manner, but without 

 suppressing the common factor, the last three coefficients of the 

 preceding cubic ; and we see the same to be true of the cubics 

 following; that is, if the criterion of imaginary roots be satisfied 

 by the last three terms of one cubic, it must be satisfied by the 

 first three of the next, and vice versa ; so that the fulfilment of 

 the condition by any two consecutive sets of three terms implies, 

 of necessity, but one pair of imaginary roots in the primitive. 



We thus arrive at the following conclusions, namely : — 



1. If the first three terms in the first cubic (or indeed in any 

 cubic) satisfy the criterion, we may infer the existence of one 

 pair of imaginary roots in the primitive equation. 



2. If the next set, the last three terms of the same cubic, also 

 satisfy the criterion, the circumstance supplies no additional in- 

 formation ; it is merely a repetition of what had already been 

 indicated, inasmuch as more than a single pair of imagina- 

 ries can never enter a cubic equation. In this case, however, 

 the following set of three (the leading terms of the next cubic) 

 must of necessity satisfy the criterion ; and so on, till we arrive at 



12 



