324 Prof. Twining on the Parailelogram of Forces. 
Art. XIV.—On the Parallelogram of Forces ; by ALEXANDER 
C. 'Twinine, Professor of Mathematics, Natural Philosophy and 
Civil Engineering in Middlebury College. 
I propose to treat of the subject in two methods. 
Method first. 
To investigate the intensity and direction of the resultant of — 
any two given forces. 
Let BA, BE (fig. 1) represent two equal forces, each of which 
call unity. Apply two new forces, BD, BC,—each being 1,—in 
such a manner that ABE, and its equal DBC, may each be a mul- 
tiple, by m, of CBE. Put z for the resultant of BC, BE, at the 
unit angle CBE, which call A. Let R, R’ represent the equal 
resultants of BA, BE and of BD, BC, which will be, respective- 
ly, in the lines BF’, BG, bisecting the angles ABE, DBC. Then 
the four forces BA, BD, BE, BC have the same resultant with 
the two, R and R’,—which, since Fig. 1. : 
FBG=EBC, will be Ry. Then zB 
Rz = Res. (BA, BC)* + Res. (BD, 
BE). But BA, BC act at the angle 
m-+-1A, and BD, BE act at m—1A. 
Therefore to find the resultant, at 
at mA by z, and deduct the resultant at m—1A. By assuming 
the value of m, successively, 1, 2,3, &c. we may find an expres 
sion for the resultant of the two equal forces, acting at any mul- 
tiple of A we choose, in terms of x and the values of m. 
Having thus shown the law of formation of the expression for 
the resultants of unit forces acting at multiples of the unit angle, : 
I shall next exhibit the law of formation of the diagonals of pat- 
allelograms under analogous circumstances. 
T resume the figure already used. But, instead of representa- 
tives of forces, let BA, BE be two sides of a parallelogram, each 
equal to 1, and having its diagonal BF, which I call D. Let 
BD, BC, in like manner, have the diagonal BG =D; and let the 
* By such expressions as Res. (BA, BC), Diag. (BA, BC), I intend the resultant 
of BA and BC, or the diagonal pertaining to BA, BC, as two sides of a parallelo- 
gram, 
