1903.)  LAMBERT—MACLAURIN’S SERIES OF EQUATIONS. 91 
ponent equal to the difference of the degrees of the first and 
third terms to the product of the coefficient of the first of 
the three terms with an exponent equal to the difference of 
the degrees of the second and third terms and the coefficient 
of the third term with an exponent equal to the difference 
of the degrees of the first and second terms shall be a large 
number. 
To illustrate the method, the following equations are 
discussed : 
(a) y& — 10y° + Gy + 1=0. 
Here all the terms are underscored, for the ratio of 
104 to 6 is large, and the ratio of 6’ to 10 is large. The 
method must be applied to (1) y— 107+ 6ry +2=0, 
{2) xy? — 10y? + 6y +x =0 and (3) ay — 10ry*?+6y+1=0. 
The computation determines the following values: 
From (1) Yo = + 3.167, — 3.167 
dy 
ae — 0.100, + 0.090 
2 
} el = — 0,008, + 0.008; 
Lo 
From (2) Yo = + 0.775, — 0.775 
dYo __ ee 
aoa + 0.107, + 0.060 
te ees 
Sige = — 0-008, + 0.016 ; 
From (3) RTE Sr ge 7 
fs = SS (Uy = — 0.007. 
dz 
The roots of the given equation are y, = + 3.05, 
Y= — 3.06, yz= +0.87, ysz=— 0.69, y;=— 0.17. 
(6) at + 4a3 — 4? — ile + 4=0. 
Here the terms to be underscored in addition to the first and 
last are probably the second and fourth, but as the ratio of 
43 to 11 is rather small, it is safer to transform the equation 
into another lacking the second term by the substitution 
x=y—1. There results 
y — 10y? + 5y + 8 = 0. 
The terms to be underscored are the first, second and last 
and the roots are obtained by applying the method to 
