Anglin — Mathematical Notes. 289 



XLYII. — Mathematical Notes. By A. H. Anglin. 

 [Read, January 12, 1880.] 



Absteact. 



I. 



If tti, oo, a^, . . . a,„ he roots of the equation 



x"' = 2h -iC"'^ + _^2-^™"" + Ps.x'"''^ 4- . . . + p,n, 

 then 



hn = Px ' hn-l + i?2 . /«»-2 + Pz • hn-S + • . • + P,n • K-m in > m), 



where A„ is the sum of the homogeneous products of ai, a,, as . . . a,^, 

 of n dimensions. 



The proof of this may be briefly indicated thus : — 

 "We have 



iij™ - px , ^'""^ - Pi . x'"''^ - ... - p,,^ = {x - ttj) (:c - tto) . . . (^x- a„,), 



"Writing - for x, ii.:-'Jtiplying off by a?'", and finally equating co- 

 efficients of x^ in both members of the equation, we shall find that 



hn - Pi . hn-l - Pi . ^„-2 - ... - P,n • 4-m = ; 



or, 



hn = Px ■ h»-i + po . h„_2 + Pi . /«»-3 + . . . + p„, . A„-„,. 



The particular case of a quadratic equation is then noticed. If 

 a, ^ be roots of oo~ = p)^ + ^, we have 



hn = P'hn-X + q-hn.i, 



where A„ is the sum of the homogeneous products of a, (3 oi n 

 dimensions. From this it is deduced that 



(2»-3)(2w-4)(2«-5) ,,^ (w+l).w , J „ 



+ v '^^ ^ g \p"'-'.q' + . . • + ' ^ ^ .pKf-' + q" 



(0 



E. I. A. PEOC, SEK. II., VOL. III. — SCIENCE. Z 



