





128 J. DOUGLAS HAMILTON DICKSON ON LEAST ROOTS OF EQUATIONS. 
For example: 
1 bs 
P RoRapeae = 1—2-—32' 4+ 72° + 52*—532° 
+ 132° +1192’ —1712°—3052° + 9892" 
+2312" — 41872” + 32632" + 134852 — 265372" 
— 274032" + 133551a"—239392"°—510265z2”" + 6060212” 
+14350392"—... 
the coefficients being connected by the relation A,,,+A,+4A, ,=0 
_ Vapt2 AR Ap ae es ee 
** tov, (A24+8A, ,Al4 16A?_,)—A,(—3A, 74455) 2 F 


and 
= MApiotA,_ + 
p cos 0 = oe = 
therefore the roots are 

=5 ee pA - 15 
Again 
2— "by 
G z a a= in 1 Gprie where the values of B,, B,... are as follows :— 
7,10, —62, 64, 244, —872, 280, 4672, —11024, —5984, 
78112, —120320, —228032, 1177984, —987776, 
— 5092352, 16111360, —1668608, —93350944, 19667.104, 
16664.10*, —151330.10*, 202677.10*, 503626.10*, —2223314.10*, 
22233140000 a, 
, ee ae 
i.e., the term in «”* is — ie 
Here 
Boy Bp 
6 oe SS yar Qn + epti =0; or B,..+2 Byii+6 B,=0 

Uaps9 go Bp—By—1Bpsa — B24 BB 4 6B? 


Dp 6? _ 4a 
fo) oe 4B; +24B,B,_, +36B;_,+2B;—12B,B,_, =6=p 
and 
ee BApiotAp _ 
p. cos 0 =  2heaee S| 
therefore the roots are , —1atiJ/5 
(V.) To complete the discussion of the determinant quadratic for the two 
least roots of F(z) = 0, it is necessary to show that it holds if the two least 
roots of F(z) = 0 be equal. 
