1903.] LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. 89 
y* —11727y + 403852 =0, 
Yor= 22.720, — 11.360 + 11.360V —3 
a Pann ty = 
da? 
2 ‘ i ane 
Sa 116, 058+ .058V—3 
2! dit, 
es Pada ze Tan 
ST gata (Ole, 010 .010Y —3 
a 
Tae — 004 — 04 
Three roots of the equation are 21.432 and 
12.444 + 19.759V—1. Applying the method to 
ay*—11727y + 40385 = 0, 
Yo = 3.4436, ie = ,0120, 4 Sh. = ,0002. 
The fourth root of the equation is 3.4558. 
This method applied to trinomial equations proves that an 
equation of degree n has n roots, determines how many roots 
are real, and presents a uniform scheme for computing all 
the roots, real and imaginary. 
II]. —GENERAL ALGEBRAIC EQUATIONS. 
The method applied to the complete equation of degree n 
(7 —1) 
. nN : Hes 
furnishes —,,—~ series, and it becomes necessary to deter- 
ow 
mine which of these series give n convergent series for the 
roots of the equation and if possible to insure rapidity of 
convergence of these n series. 
Suppose the equation of degree n to be 
AP SEIS) 1 SP oat 4 amma oA ee a oA ai 
+ i aca -- pS + OO hm -+ Bhatti -- Cr papain tee 
+ dye 4 dgyp mt +... try +90, 
and suppose the terms which are underscored to be the terms 
from which the two terms into which the factor « is not intro. 
duced must be selected by taking consecutive terms in regu- 
lar order from the left. The problem is how to recognize 
the terms which must be underscored. 
