A ROOT IN EVERY ALGEBRAIC EQUATION. 269 



which vanishes when r = 0, and finally increases without limit with r. At some value or 

 values, then, of r, we have r«,«, ... = m. Consequently, the given equation has one or more 

 roots : that is, every equation of the form x (x - a) (x - 6) ... = m has one or more roots. 



Next, it follows that if every expression of the {n - l)th degree has n - 1 roots, every 

 expression of the nth degree has n roots. First, 



ax" + bx'-^ + ... + kx + I is ax (.t?"~> + ... + k) + I, 

 which, since every expression of the (n - l)th degree has n - 1 roots, is ax {x - a) (<r - /3)... + I, 

 and this, by the preliminary theorem, has one root. Consequently, ax" + bx^'^ + ... is of 

 the form (x - a) (ax"-' + bx"-' + ...) which again is (x - a) x the product of w - l such 

 other factors. Whence oa?" + ... has n roots, if every such expression of one degree lower 

 have n - 1 roots. All the rest follows from the expression of the first degree having one 



root. 



A. DE MORGAN. 



UNrVERSITY CoLLEGK, LoNDON, 



December 18*, 1857. 



Postscript. 



I SUBJOIN some brief remarks on a couple of elementary points. 



1. I find that the following theorem is new to several mathematicians to whom I have 

 proposed it. It may be most briefly expressed as follows : Any two divergent series whatso- 

 ever, of the same character as to signs of the terms, are to one another in the ratio of their 

 last terms. That is, if a„ + a, + Og + ... + a„ and 60 + 61 + 62+ ... + 6„ give results 

 which become infinite with n, the limit of the ratio is that of a„ to b„ ; and this, whatever the 

 signs of ag, 69, &c. may be, provided only that a„ and 6„ always have like signs. Thus 

 1 + 2 + 3 ... and 1 + 3 + 5 + ... are infinites in the ratio of 1 to 2 ; and so are 1 - 2 + 3 - ... 

 and 1 - 3 + 5 - ... ; and so are 1 + 2 - 3 + 4 + 5 - 6 + ... and 1 + 3-5 + 7+9-11 +.... 

 I speak only of arithmetical summation, without reference to the value of the evolving func- 

 tion, when finite. Apply this to 2 . w* and 2 \{n + 1)*+' - «*+'}, and we have (;!;>- 1), 

 the proposition out of which Cavalieri and his successors produced a limited integral calculus. 



Apply it to 2«"' and 2|log(n + l) — logw| and we may render the connexion of l + 1 +.,.+- 



n 



and log n + const. + An "' + ... a part of elementary algebra by easy processes. And simi- 

 larly for log 1 + ... + log n and n log n — n, and generally for 20ra and ^f,^*^<pxdx. 



2. I have looked through elementary writings in vain for a classification of the species of 

 spherical triangles, as to character of sides and angles, with respect to the right angle. Ex- 

 cluding the right-angle, the cases which exist are as follows: all cases in which opposite sides 

 arid angles are of the same name ; and all others in which an odd number of acute sides is 



The substance of this paper was read to the Society on the 7th of December, as stated at the commencement. 



