324 Prof. Twining on the Parallelogram of Forces. 



Art. XIV. — On the Parallelogram of Forces ; by Alexander 

 C. Twining, 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 x 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 Fj g- j 



FBG=EBC, will be R*. Then ^^4^\ 



Kx = Res. (BA, BC)* + Res. (BD, Jk>^^^mS^^i 

 BE). But BA, BC act at the angle \/ /\ \/ 

 w+lA, and BD, BE act at m — 1A. "^^/J XZ^ 



Therefore to find the resultant, at * h g- 



the angle m+lA, of two equal forces, we multiply the resultant 

 at mL by x, 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 par- 

 allelograms under analogous circumstances. 



I 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. 



