94 Composition and Resolution of Forces, &c. 
X, Y, Z, for the first terms of the first members of (s) severally, 
and D,E, F, for those of (t), then multiply the first of (t) by cos. C’ 
and the second by —cos. B’, add the products, and we have D, cos. 
C’—E, cos. B’+R’ cos, A’(Y’ cos. C’— Z cos. B’)=0, this com- 
pared with the third of (t) gives D, cos. C’— E, cos. B/—F,, cos. A’ 
=0, (u), multiply (u) by R’ and substitute from (s), then we have 
F/X,+E,Y,—D,Z,+R(F, cos. A+E, cos. B—D, cos. C)=0,(v) ; 
this or (u), is an equation of condition that must be satisfied ; for it is 
evident that (t) are not independent equations. Again, multiply the 
first of (t) by Z, the second by —Y, the third by — X, then adding 
the products we have D,Z — E,Y — F,X=0, (w), which is the equa- 
tion of a plane passing through the origin of the coordinates, in 
which the fixed point » must be taken, it is easy to find the position 
of this plane by knowing D,E, F’,, and it is evident by (u), that the 
reaction R/ is always in this plane; hence by assuming n in this 
plane, and taking mn for the axis of z, it is evident that (t) will be 
changed to SmF(y cos. a—a cos. b)=0, SmF(z cos. a—a cos. c)+ 
R’Z cos. A’=0, SmF(y cos. c—z cos. 6)—R’Z cos. B/ =0, (x). 
We may suppose mn = Z, to be a fixed axis, then by (x) R’ 
Vv cos.? A’-+cos.? B/=R’ sin. C/=V ,E? +,F2—mn=the reaction 
of the fixed axis (at m,) in a direction perpendicular to its length; 
Leen AY 
also ee i aiXaaidl En the tangent of the angle which its direc- 
tion makes with the axis of x; the perpendicular reaction at m, is 
also easily found, and likewise by the third of (s), we have the re- 
action of the axis in the direction of its length. Again, if R=0, 
there will be but one fixed point necessary, and the applied forces 
will have a single resultant which will pass through n, and be de- 
stroyed by the reaction R’; also (u) or F,X,+E,Y,—D,Z,=0, (y) 
is an equation of condition which the applied forces must satisfy in 
order that they may have a single resultant, supposing 7 to be a fixed 
point, in the direction of the resultant. But if the applied forces 
have not a single resultant, it is evident by what has been done, that 
they may be resolved into two resultants which are not in the same 
plane, in an infinity of ways; for we have seen how to find the two 
points m and n so that their reactions R R’ destroy the applied for- 
ces, .’. they are directly opposite to the two resultants into which 
the applied forces are resolved; and it is evident that an infinite 
number of points, which destroy the applied forces like m and m can 
be assigned. 
