J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
therefore 

prey = A,Ay +2) ee (A; ra Ay—1Ay +1) = 0 
a a ; 
that is, 
w= - ae 
Dap ih a A, Ay +2 

Again, equation (6) on simplification is 
i Avag + Ay = 2 COS OAS aap, 
eliminating » , this becomes 

AZ — A,_iA, 41 
sete een Nee es AES Dl egeryt Neh is, 
Aj. — A,Ay +2 p+2 Pp p +i 
that is, 
AyAy +1 —Ap—rAv+2 _ 9 cos 6. p, 
Re pA ae 
or 
AyAps+1— Ap—rAp +2 
2 cos 0 = - 2 
J (AR — Ay 1A, 41)(Ap41 — AyAp+2) 



These two results may be written thus 
Uap + 2 Urp + 2 



LDN tape 7 EJ Uirpy 2 + Urp +e 
Uap +3 
ZCOSs0 = z ; 
E Jury +2 - Urpt4 
Note.—If 
L+ mx 
Ge Cae. ya ee 
then 
(p = 0) ee ae GA, +41 a bA, = 0, 
and 
ES pe a’ At + 2abA,A,_, + DA? _, —@A? —abA,A,_, + DAR _ 
HP Uap +2 A, + aA,A,_, + bA;_1 | < 
and 
= @WAnrot Ap oe Ap+2+ dAp ae a 
therefore the roots are 
eee rn/a2*—4b 
2b 2b 
or = 
VOL. XXVIII. PART I. pans 
127 
(7) 
(8) 
b, 
