1903. ] LAMBERT—-MACLAURIN’S SERIES OF EQUATIONS. 85 
NEW APPLICATIONS OF MACLAURIN’S SERIES 
IN THE SOLUTION OF EQUATIONS AND 
IN THE EXPANSION OF FUNCTIONS. 
BY P, A. LAMBERT, 
(Read April 3, 1903.) 
I.—INTRODUCTION. 
The modern theory of differential equations is based on 
the expansion by Maclaurin’s series of the solutions of the 
equations in infinite series. The striking analogy existing 
between the theory of algebraic equations and the theory of 
differential equations suggested the possibility of expressing 
the solutions of algebraic equations in series to be obtained 
by an application of Maclaurin’s series. After some experi- 
menting the author happened on the device of introducing 
a factor r into all the terms but two of the equation f(y) =0, 
whereby y becomes an implicit function of +. The succes- 
sive r-derivatives of y are now formed, and together with y 
are evaluated for +=0. By Maclaurin’s series the expan- 
sions of y in powers of + become known. If + be made 
unity in these expansions, the roots of f(v)=0 are found, 
provided the resulting series are convergent. 
To illustrate this method, consider the equation 
(1) yt—8y? + 75y—10000 = 0. 
Maclaurin’s series 
yy a yy es yg a 
= Yo + ae (Ot aya) + ax,33! dee ay ss 
dy PYo dy" ao, Se 
, a | 
where Yo, 5 IR Re TRY stand for the values 
f dy @y @y dy 
>a’ de®” de® dae -- when # is made zero, expands y, 
a function of x, in powers of 2. 
By introducing a factor x in the second and third terms of 
(1) an equation is formed 
(2) yt— dary? + T5ry — 10000 — 0 
which defines y as an implicit function of 2. 
