379 



are 2n dashes ; this last result being a positive number : moreover, each 

 term consists of a group of n X's. 



Now, it is plain, when in each of the terms, or individual groups, 

 entering a derived equation, the number of dashes is odd, that X must, 

 of necessity, enter that group once, or some odd number of times ; and that 

 when the dashes in each group of X's, in a derived equation, are even 

 in number — since that even number can be made up without any X t 

 all — there must be one group from which X' is absent. It follows, 

 therefore, that the root of the simple equation X' = 0 is equally a root of 

 every ascending equation of an odd degree ; but that for this, and for 

 all real values of every one of the functions of even|degree is posi- 

 tive ; hence, if we diminish each of the roots of the proposed equation 

 by the root of X' = 0, we shall arrive at a transformed equation in 

 which the alternate terms will be zeros, each zero being between plus 

 signs. 



In diminishing by the root of X = 0, should it consist of two or 

 more figures, Newton's Eule, in the case here considered, is of special 

 service ; as the conditions of imaginarity are likely to hold^and fail al- 

 ternately before the full development of the root of X' - 0 is completed. 

 And the same may be said in reference to equations made up of qua- 

 dratic factors furnishing imaginary pairs not strictly identical, but only 

 nearly so. But in every application of this rule to the determination of 

 the number of imaginary pairs in an assigned interval [a, £>], care must 

 be taken that pairs outside that interval are not included in the enume- 

 ration ; that is, that the evanescencies — which Newton's Eule anticipates 

 ■ — are not any of them delayed beyond the limit b\ such as are so delayed 

 imply pairs belonging, of course, to the succeeding interval [b, e]. 



Note 2. 



It may be well, before closing this paper, to give a short practical 

 illustration of one or two of the theoretical principles established in 

 the latter articles of it. The equation of the fourth degree, at p. 368, 

 is well adapted to this purpose, since the roots are all doubtful, and all 

 lie within the narrow limits [0, 1]. 



The inquiry is — Does an imaginary pair enter this equation ? and 

 if so, by which triad of terms is the entrance of the pair first indicated ? 

 It cannot be by the leading triad, from the principle at (27) ; that 

 is, an imaginary pair cannot enter the derived equation X 2 = 0. Can 

 a pair enter the derived equation of next higher degree-— namely, 

 X, = 0? 



The roots of the derived quadratic X 2 = 0 — that is, of 8x* - 10% 

 + 3 = 0 — are readily found to be -5, and *75. Taking the smaller of 

 these as a transforming number, and working up to X 0 , if imaginarity 



