(538 DR WALLACE ON A FUNCTIONAL EQUATION. 



ter?ection of the directions of the other two. Draw the vertical line OK, and 

 from C and D draw perpendiculars CL, DM. Put (p, cp' . <p" for the angles LCK, 

 LCD (or its equal CDM), and KDM, these being the angles which the links BC, 

 CD, DE make with the plane of the horizon. 



By the nature of the centre of gravity, and similar triangles, 



.:.' = DO:CO = DM:0L=J^:^l^=M.:.|2_. 

 Now ?^T~ ~ 7^ T~ ~ ^4~ =tanKCL— tanOOL=tan0— tan0'; 



^"*^ DM""D"M ~ j)-i^- = tanODM-tanKDM = tan<^'-tan^": 



Therefore, m:m= tan cp — tan (^' : tan (p' — tan (p" . 



Hence we have this proposition : 



Theorem. — In an equilibrated polygon, the loads on any two joints arepropor- 

 tional to the difference of the tangents of the angles (p, <p' winch the sides about that 

 joint make with the plane of the horizon : And ifcbe put to denote some constant 

 fm'ce or pressure, 



tan ^— tan ^' = — . (1) 



This is the condition requh-ed. 



Corollary. — Hence, if the angle which any one of the rods makes with the 

 horizon be given, the like angles which all the others make will also be known. 



25. By a theorem in the calculus of sines, 



i /A. i>\ tan</)-tan(^' 

 tan (d) -</)') = - — -^--r-j. — ^ ; 



therefore, tan <p - tan <p' = tan {cp - (p') (1 + tan <p tan (p'). 



Hence, in the equilibrated polygon, m and c representing the things ah-eady spe- 

 cified, 



— = tan {(p - (p') (I + VAn(pidM<p') (2) 



, If the number of links of the chain be very great, so that the angle made by 

 any two adjoining links is very obtuse, the tangent of the difference of the angles 

 will be almost proportional to the difference of the angles themselves ; and the 

 product of the tangents will be almost the square of either. In this case, 

 \ + \.a.u(p tan (p' = \ + tan'' <p = sec" (p, 



m 



and —={cp-(j)')sec'^cp nearly. 



c 



When the number of links is infinitely great, so that the figure which they form 

 may be considered as a curve, cp-cp' is d 4), the diflerential of <p and 



-. scc^ (p d(p = d {tan (p) (3) 



m 

 c 



