



130 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS, 
gives its least root ; and that the equation 
a ye 5 1 —0 
ease Ay , Any 
aN Aya; 15 
gives the two least roots, when p is infinite. It will now be shown that the 
equation which gives the three least roots is 
Let the coefficients of this equation be respectively ts)13, Wspi2, Uspir» Usp» 
as those of the quadratic are 2,49, UWp41, Us. Then evidently 
Usp43 = Ay Uapz2 — Apa Uopys + Apo Wrpss 
Uszprg = fXp1 Urpr2 — fhpe Uopizg + aa Ups ‘ 
Usp41 = Anse Urp_2. — Ags Up 1 + ake Uap 
Us = Apt Urp—2 A, Uy a + Ay 4 Urp 
(It may be noticed that the values of these coefficients are not all symmetrical ; 
the reason is, that in order to express the w’s of the third order in terms of 
those already formed of the second order, the column of multipliers has to be 
so chosen as to bring in the w’s of the second order; otherwise terms of the — 
forms A’,—A, »A,42, A,A,i1—A,»A,+; would occur, which have not yet been 
considered, though each is homogeneous in terms of the least roots and of the 
orders 2p, 29 +1 respectively.) 
The calculation of these coefficients may now be performed—in the light of 
what has already been done—by considering only the terms which contain the 
least roots. Let 7, s, ¢ be the three least roots—then 


> g(r) 1  g(s).e@) (s—H¢—8) >( g(r) g(s).e(¢) (s—t)¢—s)(s +2) 
Usp = ee 7? ¥(s). F(t) (stp ONE Get he FOF @ (st)P+1 
o(r) 1 9(s).9() (6—A(t—s)st 
+P) FF FOLO  GirA 


_ 9(7).9(s)@) (s—A(t—s) , 9(S)Po(t) @—At- 9(s).eO) (s—t)(¢—s) 
“FOFOFO Pe) TFOR® wT tPErOP ene + Oe 


