90  LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. [April3, 
If the factor x isomitted from the first two underscored terms 
a 
Y= (— 1 *. if from the second and third underscored terms 
a 
we) 
1 
yo=(—2)"; if from the last two underscored terms 
- if from the third and fourth underscored terms 
1 
Y= (— ‘ Lia Altogether n values of y, are found, and it 
is seen ata glance what values of y, are real and what are 
imaginary. In order that these values of y, shall be close 
approximations of the roots of the given equation, the suc- 
Tyo BYo VY UY |. 
4 s . € 
cessive derivatives a : 
Berens 1 «6X? aS dat 
- must be small. 
dx» 
2 
Forming ove corresponding to y,= (— >) and assuming that c 
Ly a 
is of such a magnitude that the term containing c overshadows 
all the other terms in the numerator of = it is found that 
0 
Ye is necessarily small if the ratio of b**+' to a'c* is numeri- 
i) 
cally large. This same condition insures that the following 
BY PYo 
Ota ue as 
In like manner it is shown that the derivatives correspond- 
1 
derivatives . « are small. 
ing to I,=(— 5) are small provided the ratio of c!+™ to 
4=d@' is numerically large, and that the derivatives corre- 
1 
sponding to y,= (—2) are small provided the ratio of d=—*— 
to cmsn—k-l-m is numerically large. This ratio should, if 
possible, be made larger than 10 to insure rapid convergence. 
The directions for underscoring terms are therefore as 
follows : 
Underscore the first and last terms of the equation. Such 
other terms are to be underscored as satisfy the condition 
that if any three consecutive underscored terms be chosen, 
the ratio of the coefficient of the middle term with an ex- 
