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 y 

 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 



DF 



other. Therefore N-fN=DF, or N=-q-= 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. 



Remark. — The foregoing method of deriving the diagonal per- 

 taining to multiple angles, from the diagonal at the unit angle, 

 leads, demonstrably, to the equation 2cos. wA=f- mx m ~ 2 -{- 



m — 3 

 m. — rj— / m-4 , &c. in which A is any given angle, m any given 



entire number, and x twice the cosine of A, — the series being 

 supposed to end with the term in which the exponent of x be- 

 comes 1, or 0; or, otherwise, to the equation 2 cos. mA=q>(m,x) 

 -f<r( — m ^x\ in which y(m,x) designates the entire series above 

 given, without limit, and m is unlimited in value, — which equa- 

 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. 

 Problem. Knowing, in intensity, the resultant of two equal 

 forces, to investigate that of any two forces. 



