6 PHYSICS. 
desired to divide this into two others, of which one, AD, shall be given in 
intensity and direction; then the other force wil] be found in intensity and 
direction by the third side, CD, of the triangle ACD. Draw, for instance, 
AB parallel and equal to CD, then AB and AD will form two sides of the 
parallelogram of forces, whose diagonal is the given mean force, AC, this 
being the resultant of the two forces AB and AD, determined in intensity 
and direction. If neither of the lateral forces be given in intensity and 
direction, then the first might be assumed at pleasure. 
When three forces, AB, AC, AD (fig. 4), act upon a body, the resultant 
of the first two may be found, then that of this resultant and the remaining 
force. The diagonal, AG, proceeding from A, will be that of a parallelopi- 
pedon, which may be constructed from the edges, AB, AC, AD. This 
parallelopipedon is called the parallelopipedon of forces, by means of which 
it becomes possible to determine the direction and intensity of the mean 
force, when the three forces, AB, AC, AD, do not lie in the same plane. In 
this case, supposing AB, AC, AD, to be projections of these forces, then the 
line AG will be the projection of the diagonal of the parallelopipedon formed 
on these three lines—in other words, the projection of the resultant of the 
three forces; and in the theory of projection we have already learned 
how from the projection of a line to obtain its true size and direction. 
The mean force of three or more forces acting together on a body, is 
found by the simple construction in fig. 3. From the extremity, B, of the 
line AB, representing one of these forces (any one being taken indifferently), 
draw a line, BC’”, parallel and equal to the second force, AC; from C’”, a 
line. C’"D’”’, parallel and equal to the third force, AD; from D’” the line 
DE", parallel and equal to the fourth force, AK. The line AE”, drawn 
to the extremity of the last of these parallels, will be the mean force required. 
That the Jine Ac is, in magnitude and direction, the general resultant, is a 
consequence of the fact that, when the parallelograms of forces, ABB’B”, 
ACC'C”, ADD'D”, AEE’E”, are constructed on this mean force, the single 
forces, AB”+AC”+AD"+AE"=AE", and that all the parallelograms have 
a common side in the line B’E’. 
An equilibrium between three forces must occur whenever any two of the 
forces are equal and opposite to the third. The proposition of the parallelo- 
gram of forces can be exhibited practically. Let, in jig. 15, the pomts A 
and B be fixed pulleys, in the same vertical plane, over which is passed a 
string. Let now the weight, W, act on one end of the string, W’’ on the 
other, and W’ between the two, then all will be in equilibrium in any one 
position of the string. Three forces are now acting upon the three points, 
A,B,C, in the directions CA, CB, and CW’. It can be readily shown whe- 
ther the law of the parallelogram has its application here. Suppose, now, 
that W= 2 lbs., W’’= 3 lbs., the question becomes, what must be the magni- 
tude of W' when the angle ACB is, for example, = 120°. Construct a 
parallelogram of which one side = 2, the other = 3, and the angle included’ 
between the two = 120°, and find the diagonal about = 2?, making the 
weight of W’= 2? lbs.; then the angle ACB, made by the string, will be 
= 120°. DB represents the amount of the foree W”, AE that of W, and 
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