446 Trcmsactions. — Miscellaneous. 



If, now, y be so chosen that -^ =t "^j we have 



a;Cos^ — yCoscf) — - xCosO — y xCoscft. 



= fq(pqCos6-rsGos4,). 



Hence, FT --=d-p+^(2f + (/—f--s"-), which is independent 



of the angles 6 and ^, and retains therefore the same value, 

 however the links be turned about their joints. 



By taking S suitably, this value of FT may be made any- 

 thing we please. If, for instance, — 



r>q{d-p) 



FT becomes Ud-2j) or ^DP. 



Hence, if FO be taken equal to FD, OT will also equal TP. 

 It follows that EO must equal ED, and SO must equal SP. 



Thus, if to the original framework there be attached two 

 other links at the points E and S, equal respectively to ED 

 and SP, and these links be hinged together at O, the point O 

 must always lie somewhere in the straight line AD. 



Thus, if APD be lixed, and the other links be moved 

 any possible way, the point O will describe the straight line 

 DPA. 



In the particular case which the model illustrates the 

 points E and B coincide, so that a =- s; also q and r are equal, 

 and therefore also b and c. The figures being similar, we 

 have in general — 



__ _ ^ c_^d 



p q r s ' 



Hence, in this particular case jjcZ = as =^ s^, and 



Hence the point S must coincide with Q. 



Also, II = X— =zx—^=q—=b; so that E coincides with C. 



' "^ pq P -'^ P 



