On the Parallelogram of Forces. 305 
would do by the action of some single force in that direction; the 
single force is called the resultant of the applied forces, and its di- 
rection is that of the resultant ; also each of the applied forces is said 
to be a component of the resultant. It is evident that if the resultant 
is applied to the point in the opposite direction, it will balance the 
components; and that it will produce the same effect on the point in 
any direction, as its components ; therefore the resultant may be sub- 
stituted for the components, and reciprocally, the components for the 
resultant in any calculation. If two forces are applied to the point 
in the same direction, their resultant evidently equals their sum, but 
if in opposite directions it equals their difference ; if the directions of 
the forces form an angle, the resultant will manifestly be in the same 
plane with the components, and its direction will be intermediate be- 
tween their directions ; if the components are equal, the direction of 
the resultant will obviously bisect the angle formed by their direc- 
tions. 
We will now proceed to determine the direction and quantity of 
the resultant. Suppose then, that two forces 2 and y, whose direc- 
tions form a right angle, are at once applied to a material point M ; 
to determine the direction and quantity of their resultant. 
Put P=3.14159 etc. = the semi-circumference of a circle whose 
radius =1; let z denote the resultant, 4 the angle which its direc- 
tion makes with that of x, then soe the angle which its direction 
makes with that of y. 
If x and y are changed to nx and ny, it is evident z will become 
nz, and that 4 will not be changed, or if " is invariable, 4 will be in- 
Z 
. ~~ av ° . . . 
variable ; but if — varies, 4 must vary ; reciprocally, if 4 varies, ~ 
z z 
: i 
must vary : hence the relation between — and 4, may be expressed 
z 
by -=0('); (1), also by changing é into anh and «into y, we have 
P 
tao(-—«), (2); where (4) denotes a function of ¢, whose form 
zi 
will manifestly remain the same, however é may vary. Again, we 
may suppose 2 to be the resultant of two equal forces R and S; R 
acting in the direction of 2, and S in the plane of x and y, its direc- 
tion being on the same side of w with that of z, and making an angle 
