347 



the primitive equation; but the converse of this is not true, except 

 under the peculiar circumstances noticed above ; that is, except when 

 the original coefficients satisfy one or more of the conditions [3]. It 

 is exclusively with this class of imaginary roots that we propose at pre- 

 sent to deal : we shall have nothing to do — at least till notice is given 



with any of the imaginary roots of an equation, the existence of which 

 roots is not discoverable from the mere coefficients of that equation 

 when submitted to the tests of imaginarity marked [3] or [5] at 

 page 344. 



It may be well to give a specific name to pairs of imaginary roots 

 of this class: we shall call them primary pairs; and the conditions 

 [3] or [5], by which their entrance into an equation is discovered, 

 are merely the embodiment in formulas of these verbal statements, 

 namely : 



For a primary pair to exist, either, 1st, the second term of the pro- 

 posed equation must vanish (for the proper transformation) between 

 like signs; or, 2nd, the second term in the reciprocal of the pro- 

 posed, or of some one or more of the derived equations, must vanish 

 between like signs. 



Of course, as already proved at (7), when the second term in the 

 direct primitive vanishes between like signs, that is, when the condition 

 [1] has place, the second term will also vanish from every derived 

 equation, down to the quadratic, inclusive. But the condition [2], 

 supplied by the final triad of terms in the primitive, will not be trans- 

 mitted to the first derived equation, since the last term A 0 will not 

 enter that equation ; but whatever intermediate triads in the primitive 

 satisfy the conditions [3], these triads, one after another, must become 

 final triads eventually, in some of the derived equations ; and the con- 

 ditions [3] being satisfied for the corresponding triads in the primitive, 

 they" must be satisfied for these also, as proved at (7). The second 

 term, therefore, in the reciprocal of every such derived equation must 

 vanish between like signs. 



(10) Such are the peculiar circumstances exclusively and invariably 

 attendant upon the entrance of primary pairs of imaginary roots into an 

 equation. As to the number of such pairs, when two or more of the 

 conditions [3] are satisfied, or as to whether more than one pair can 

 ever be safely inferred, however many of these conditions are satisfied — 

 these are points in reference to which nothing can be determined at 

 this stage of the inquiry, except the fact that in a cubic equation, 

 though both the triads furnished by its four terms should equally satisfy 

 the condition of primary pairs, yet only one pair of imaginary roots can 

 enter the equation. A single illustration of the fact, in an equation of 

 higher degree than the third, will suffice to show that the fulfilment of 

 consecutive conditions [3], how many soever, does not imply, of neces- 

 sity, more than one pair of imaginary roots. Thus, take the equation 



4£ 4 - 9x 3 + 8x 2 - \x + 8 = 0, 



