542 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
IV. It is noticed in what has gone before that, of the two matrices employed 
in any symbol of roots, the one refers more particularly to powers (and their 
sums) of roots, while the other refers to products of roots. Hence, using the 
principle of multiplication of matrices, I write, for example, the equation 
eos oe 1 : 1 : 1 ; ai 
a,p,y a B+tyt8S ,at+ty+8 ,a+B+8 ,at+B+y iF} 
a’ ,B’,y , 8 | By+Pdt+ys, ay+ad+y6 , a8+ad+4+ Bd , aB+ay+By 
a Rapp CVE. Sw On” ga og eh BOR wee ae cen, | 
L dae 
=| 83P;—s,P.+5,P;— s)P, , 83P,—sP3+5,.P, , 5;3P;—s.P, , s3P,|, 
s,P, —s,P, + s)P3 , SP, =s PP, +P. Sr sik: 5 Se 
| $P, — sP2 , $P2—sPs , 4P3—%P, , Py 
Py , SP» » SPs » SPs 
14 Py ncaa] oa ae jideeat 
—where s,—s,1P,+5,.P;— ...+5P,=0,and s,=sum of mth powers of 
the roots of the equation, except when 2=0, in which case s, is the same as the 
suffix of the P with which it is associated—in the symbolical form 
Syl es 
AY 22.185. Soe Ses. Soll Wenn teen ele ele Om 
3) SeSica tee | Pe pera eas 
FAN 2 Si: RES orate | Boy ie ; 
1/3 Sic Us EP Oe 
Q) PL i eS 
Example : take, in the case of a biquadratic, 
pl 
(7 (OPS 8, S55 Sey Sos 82, S15 Soll bay —Ee, Ps, —Pil .. ae 
AL Sig Gig Ba 5 Si ee eee ee 
3|2 1S. 5815 80y- 5+ 9+ 9s Py c= Ease 5 
ae 5 ae | eae er 
=| s.P,—s;P.+ 5,P;—5;P, , 56P,—8,P,+5,P, , S65Ps—s5Ps , s6Pa|. (11’). 
8,P,—8,P_+ 5.P3—5,P, , 5.P2.—5;P3+5.P, , 54Ps—53P4 , Ps 
s,P,—s,P,+ 5 Ps , %&P,—5,P3+5 Py, :P3s—4Py , 52P, 
