370 Prof. Young on the Completion of the 



these latter from the former, adding in 5 at each reverse step, that 

 particular constant (or final term) which, in the direct step, was 

 made to disappear. 



In the equation of the fourth degree (the result of the first 

 step from C) there enters an imaginary pair necessarily and ex- 

 clusively dependent for its imaginary character on the pair in C. 

 But the pair in C is not dependent on this pah- in the biqua- 

 dratic; if it were, it would be dependent on the pair in C, 

 which it is not. In like manner, the equation of the fifth degree 

 in the next reverse step has an imaginary pair dependent on 

 the before-mentioned pair in the preceding result, and there- 

 fore on the pair in C. The pair in C is therefore equally in- 

 dependent of this pair, and so on through all the reverse steps 

 up to R; that is, there is a pair of imaginary roots in R of 

 which the imaginary pair in the cubic C, derived from R', 

 is independent. But imaginary roots can enter R' only as a 

 consequence of imaginary roots entering R; and imaginary 

 roots can enter C only as a consequence of imaginary roots 

 entering R', and therefore only as a consequence of imaginary 

 roots entering R. But it has been shown that the imaginaries 

 in C do not enter as a consequence of that particular pair in R 

 of which the pair in C (in the reverse process of derivation) is 

 the source ; hence the pair in C must be the consequence of 

 some other pair in R, which therefore has at least two pairs of 

 imaginary roots. Consequently the primitive equation has at 

 least two pairs of imaginary roots. 



In the foregoing reasoning R is regarded as derived from C 

 (after restoring the factor previously expunged) by the process of 

 integration ; and it is to be observed that it is the general inte- 

 gral at each step that owes a pair of imaginary roots to the en- 

 trance of a pair into C, — -in other words, that though the constant 

 completing any integral be quite arbitrary, or even zero, a pair 

 of imaginary roots, resulting from the pair in C, must still enter. 

 The constants actually introduced are each of assigned value, as 

 a specific equation (R) with assigned coefficients is to be deduced : 

 these constants, added one after another as the derivation proceeds, 

 may cause the introduction of additional imaginary pairs into 

 the successive equations of even degree ; but none of these pairs 

 are traceable to the pair in C, and a pair of imaginaries trace- 

 able to those in C would still enter each of the ascending equa- 

 tions, though no constants at all were introduced. 



I have felt anxious that my demonstration of Newton's Rule 

 should be thus rendered complete, not because I regard the Rule 

 itself as of much practical importance in the solution of nume- 

 rical equations, but because I have always thought the line of 



