a 
326 Prof. Twining on the Parallelogram of Forces. 
a difference, may evidently be done. Now, by what has been 
shown, the resultant of the equal forces, acting at the angle nz, 
is represented by the diagonal pertaining to the sides with the in- 
cluded angle nz’; and thus—since one of these multiples has a 
ratio to two right angles of greater inequality, and the other of 
less—a case is constituted in which it appears that the resultant 
of two equal forces bisects, not their interior, but their exterior 
angle; which is absurd. Therefore ABC and DBE cannot differ; 
and it is made evident that the resultant of any two equal forces 
is represented, in direction and intensity, by the diagonal of a 
parallelogram whose sides which include the bisected angle rep- 
resent, in direction and intensity, the forces. 
Let then BA, BE have their resultant BF. Let N represent 
the entire effect of each, resolved in the direction BF. The only 
residual effects must be normal to BF, and must destroy each 
other. "Therefore N+N=DF, or hea cos. ABF’, to the ra- 
dius BA. By the same conclusion the residual effect must equal 
the cosine of the complement, that is to say, the sine of the same 
angle. Therefore a force represented, in direction and intensity, 
by the hypothenuse of a right angled triangle is the resultant of 
the forces represented, in the same respects, by the other two. 
sides. And from this the law that regulates the resultant of for- 
ces acting at any angle whatever is a deduction so obvious that 
it need not, here, be considered. 
emark.—The foregoing method of deriving the diagonal pet 
taining to multiple angles, from the diagonal at the unit angle, 
leads, demonstrably, to the equation 2cos. mA=z"—mz"? + 
m.—>—x"-*, &c. in which A is any given-angle, m any givel 
entire number, and y twice the cosine of A,—the series being 
supposed to end with the term in which the exponent of % be- 
comes 1, or 0; or, otherwise, to the equation 2cos. mA=9(™; 4) 
+9(—m, z), in which 9(m, 7) designates the entire series above 
given, without limit, and m is unlimited in value,—which eque 
tions, it is well known, are of signal use in the discussion and 
treatment of certain circular functions. Another application of 
the same principle of investigation, not necessary to my subject, 
but collateral with it, and worthy, it may be, of notice, I subjoin- 
roblem. Knowing, in intensity, the resultant of two equal 
forces, to investigate that of any two forces, 
