EXPOSITION OF THEORY OF POWER SERIES. 217 
whence the equations of motion in this case assume the very simple 
form — 
M n*x l = 
+ 
L ^1\ 
p t dx x ) 
M 7l n 
&c., 
Pn (fn + 
Jl ^ \ 
Pn ' dX H f 
It seems hardly possible to have the equations of motion in a more 
concise form than these, inasmuch as they give explicitly the accelera- 
tions of the co-ordinates. 
If there be no external forces, then/) , •• f n are all zero, and the 
equations become — 
M n x x l =■ 
dT 
dx x 
M 7l*X n — 
dT 
dx n 
We note that the equations — 
dT 
dx l 
= 0 . 
dT 
dx n 
0 
express the conditions that the movement shall be such that there is 
no immediate tendency to vary it. 
5. — AN ELEMENTARY EXPOSITION OF THE THEORY OF 
POWER SERIES. 
By G. FLEURI , Licencid es -sciences mathtmatiques , LicencU is- sciences ■physiques. 
The representation of functions of one variable by means of series 
(Power Series) plays a very important part in mathematical science. 
However, with the exception of ChrystaPs Algebra — a highly valuable 
book (see especially Part II., Edinburgh, 1889) — there is not a 
single English treatise dealing with the subject in a vigorous manner. 
Moreover, Mr. ChrystaPs work is incomplete, mainly owing to the 
fact that he does not make use of the notions of Differential and 
Integral. 
I have attempted, in the following, to give an exposition — as 
elementary as possible — of the important theory in question ; but, 
in order to preserve uniformity and simplicity in the demonstrations, 
I have had to restrain the generalitv of the main theorem— that of 
Abel. 
Throughout this paper I use the word “ series ” as meaning a 
succession of an infinite number of quantities following a definite law 
of formation and connected by the signs plus or minus. 
The quantities with their signs are called the terms of the series. 
