86 Composition and Resolution of Forces, &c. 
that it may be destroyed by the reaction. Let X, Y, Z, denote the 
sum of the components of all the forces which affect M, when resol- 
ved in the directions of , a z severally ; Let N denote the reaction 
of the surface, p= V/ (x —d) yee (2 (ZF)? =the perpendic- 
ular to the surface drawn through M, d, e, f bemg the coordinates of 
the origin of p; we shall suppose that the origin of p is taken on 
that side of the surface, towards which N is directed. 
d—«x 
Now by resolving N in the directions of « y z, we have N x ’ 
z 
» for the values of N when reduced to those di- 
NX i d, Nx is 
iw iY 
rections; hence for the equilibrium of M, we have X+-N x =0, 
e-y Dee 
Y+Nx— =0, Z+NX me (15). By the nature of the 
Jaa 
du 
(16), and by (14) Tae ly + ydz=0, (17), multiply (17) by 
the indeterminate L, add the product to (16), then put the coefti- 
cients of the ce ae da, di dz separately, =0, and we have 
RTs fy dit 
—+L7=0, 5 L7,=9, (8); hence L= 
pen / (2. (=) ac an aie 5 *. denoting this value of LL, when 
multiplied by N, by N’, we shall have by (18) and (15), en 
—d 
perpendicular, we have P i bs We 
du du du hh 
NW =0, mate Z+N’7_=0, (19); by elimmating N‘ 
du u du 
from (19) we have x5 aie: =O, ee Z7=9, (20), for the 
conditions of the waa of M; (14) and (20) are sufficient to 
find where M must be placed on the given surface, to be in equilib- 
rium, and by (19) we have N=V X2+Y2+Z2 =the force with 
which the surface must react in order to destroy the resultant of the 
applied forces. If Mis to be m equilibrium on a line, which is 
formed by the mutual intersection of two surfaces, which are deno- 
ted by u=0, w’==0, (21); then by using the same notation as be- 
fore, and puting L/ pei +(5,) +(e] : N=t0 
