Motion of a System of Bodies. — 289 
ad? y —yd*z 
eT at 
spectively, then add the products, and we have 
e'd2y! —y/d? x! 
dt? 
Py, also = Q’z' — P’y’, and so on for all the bodies M, 
m’, &c. Let a-unit of m act on a unit of m’ with any force p, then 
(by the well known law of equal action and reaction ;) a unit of m/ 
will react on a unit of m, with the force —p, which is directly oppo- 
site top; hence mp= the whole force with which m acts on a unit 
of m’, and —m/p= the whole force of the consequent reaction of m’ 
on a unit of m; let f denote the straight line which joins m and m’, 
then evidently the forces mp, —m/’p are exerted along the line f: 
hence by resolving mp in the directions of the axes of z and y, we 
Seis, 3 nee 
have (=) mp and (ee 
P’ and Q’ which arise from the force mp, and — (=) Xm'p, — 
respectively for the parts of 
teat at) x m/p, are the parts of P and Q which arise from the force 
—m'p ; hence, (for simplicity,) considering these oe, only, we 
have (Qe—Py)=( (5 ly-(").) am! -(4> 2) m'p also 
Z lithe. sas Me 
: — 7! 
Q7'—- P= - (= ¥ =) mp, ewe have m(Qz—Py)-+-m'(Q’z’— 
P’y’)=0: it is hence evident that if we multiply the equations 
d? d? chs ‘d?x 
x - t nd =Qz— | fe sr ages lei Z &e. by mM, m’, 
—yd2x 
&c. respectively, and add the products, we shall have Sm ee) 
= Sm(Qz— Py) which is independent of the reciprocal actions of the 
bodies on each other, for the mutual actions of every two of them 
will destroy each other as above; the equation which we have ob- 
tained is the same as the first of (18), and the second and third of 
(18) are easily found by a similar process. 
(To be continued.) . 
Vou. XXV.—No. 2. 37 
