[ 369 ] 



LIII. On the Completion of the Demonstration of Newton's Rule, 

 and on a general property of derived Polynomials. By J. R. 

 Young , formerly Professor of Mathematics in Belfast College*. 



PRESUMING that the readers of the present communication 

 -*- are familiar with my papers on Newton's Rule in the 

 Magazines for August and October last, I now proceed to sup- 

 ply what appears to be necessary in order to complete the de- 

 monstration of that rule. And first I observe that whenever 

 Newton's condition of imaginary roots holds for any triad of 

 terms in a primitive equation, it necessarily holds also in every 

 equation derived from it into which the same triad of coefficients 

 enters ; and consequently it holds for every reciprocal of these 

 derived equations. 



This is a circumstance peculiar to the class of imaginaries 

 thus indicated; and from which it follows that the cubics derived 

 from the reciprocal equations will retain among them just as 

 many several conditions of imaginary roots as are presented to 

 us in the primitive equation. This truth is sufficiently esta- 

 blished by the series of cubics, in a general form, deduced from 

 the equally general reciprocal equations. 



Taking these cubics in order, from the first to the last, we see 

 that the entrance of an imaginary pair into an advanced cubic 

 may be necessitated by, and therefore dependent on, the entrance 

 of an imaginary pair into the preceding cubic just passed over ; 

 and that this again may necessitate the entrance of a pair into the 

 next succeeding cubic, and so on. 



Suppose, however, that these dependencies cease, and that we 

 at length arrive at a cubic the imaginary pair in which is unde- 

 rived from, and independent of, preceding pairs. Would such 

 an occurrence warrant the inference that there must be at least 

 two pairs of imaginary roots in the primitive equation ? That it 

 would, is the principle assumed in my early investigation of 

 Newton's Rule, it having presented itself to me at the time as 

 an axiomatic truth : it may be proved to be a truth as follows : — 



Let there be taken the cubic (C f ) in which the imaginary pair 

 does not derive its character, as an imaginary pair, from any 

 preceding cubic, and also one of these preceding cubics (C) 

 itself; then representing this preceding imaginary pair by I, 

 and the other pair by 1', we know that the entrance of I' is not 

 a consequence of the entrance of I, but is entirely independent 

 of the entrance of I. 



Calling the two reciprocal equations from which the cubics in 

 question (C, C) have been derived R, R', we may, by reversing 

 the process by which C, C were deduced from R, R', deduce 



* Communicated by the Author. 



