Composition and Resolution of Forces, &c. 85 
Art. VIL—Of the Composition and Resolution of Forces, and 
Statical Equilibrium; by Prof. Turzopore Strone. 
Continued from Vol. xxvi, p. 310. 
Let us resume (9), and suppose that «x, y, 2, are the rectangular 
coordinates of the material point M, when referred to three fixed rec- 
tangular axes drawn through any given point; then (9) will exist as 
before. Imagine planes to be drawn through the directions of R, r, 
r’, &c. at right angles to the plane z, y and let ,A, ,a, ,,a, &c. denote 
the angles which their lines of intersection with the plane a, y, sever- 
ally make with the axis of x. ‘Then it is evident that cos. A=sin. 
Cos. ,A, cos. a=sin. ¢ cos. ,a, &c. cos. B=sin. C sin. ,A, cos. 6 
= ieesm.d, cc.5 put ism ©— i, 7sm. ¢—t,74sm.c—¢, 
&c. .". Reos. C=T cot. C, rcos.c=tcot. c, &c.; substitute these 
values in (9), and they will be changed to ¢ cos. ,a+-?’ cos. ,,a-+ &e, 
=T cos. ,A, ¢sin.,a+?¢/sin. ,a+ &c.=T sin. ,A, tcot.c+t’ cot. c/ 
+ &c. =T cot. C, (11); where it is evident, that T, ¢, &c. are the 
values of R,r, &c. when resolved in a direction parallel to the plane 
x,y, and that T cot. C, tcot.c, &c. are the values of the same quan- 
tities when resolved in a direction perpendicular to the same plane. 
To the first of (11) multiplied by y, add the second multiplied by. 
—zx, and we have t (ycos. ,4 —asin. ,a)+¢/(y cos. ,,a—asin. ,,a) + 
&c. =T (ycos. ,A—asin.,A), (12); draw from M a perpendicular 
to the axis of z, and from their intersection draw the perpendiculars 
P, p, p’, &c. to the directions of T, ¢, t’, &c.; then we shall evident- 
lv have ycos. ,A—a#sin. ,A =P, ycos. a—a sin. a=p, &c. hence 
(12) becomes ¢tp+t/ p’+ &c. = TP, (13). If we suppose the 
force ¢, to be applied to the extremity of p, it will tend to turn it 
around the axis of z; tp is called the moment of the force r, re- 
lative to the axis of z; hence by (13) the sum of the moments of 
the components r, 7’, &c., relative to any axis, equals the moment 
of the resultant relative to the same axis: Mec. Cel. Vol. I, pp. 
12, 13. 
We will now suppose that M is pressed by the forces against any 
given surface, whose equation is u=0, (14), w being a given func- 
tion of the coordinates «, y, z, which determine the position of M ; 
to find the conditions of the equilibrium of M upon the surface. 
It is evident that M must be pressed by the forces against the 
surface so that their resultant may be at right angles to it, in order 
