Aldis. — To describe a Straight Jbine by Link-icorh. 445 



As the bar OP does not move it may be dispensed with, 

 provided the points O and P are fixed at a distance from one 

 another equal to OQ. Thus this mechanism requires only 

 five bars, instead of the seven involved in the Peaucellier 

 cell. 



Another arrangement of links, by which a very long swing 

 in a straight line can be obtained, requkes some j)i'eliminary 

 geometrical explanation. 



Let ABCD be any quadrilateral, and APQE a second 

 having its sides AP AE coincident in direction with AD 

 and AB. Let also the lengths of the hues AP, PQ, QR, EA 

 be in the same ratios to one another as the lengths AB, BC, 

 CD, DA. Then it follows that the figures ABCD and APQE 

 W'ill be similar, and if the lines represent rods jointed at all the 

 points of meeting the angles AEQ and ADC will always 

 remain equal in whatever manner the rods are tm-ned about 

 their joints. 



We shall now prove that, if S be any point in PQ, a point 

 E can be found in DC such that when ST and EF are 

 drawn perpendicular to AD the length FT will remain invari- 

 able in whatever manner the links are turned about their 

 hinges. 



Let the lengths of AB, BC, CD, DA be denoted by a, b, c, 

 d, and those of AP, PQ, QE, EA by jj, q, r, s. Also, let the 

 angle ABC or APQ be called 0, and the angle AEQ or ADC be 

 called cj). 



Then, joining AQ, by a well-known trigonometrical theo- 

 rem — 



AQ2 =p''-\-q''-22jqCose =- r-i-s-'-2rsCos<^. 



.-. ^jgCos^— 7-sCos^ = IQj^r/ — r'- — S-). 



Let, now, PS = x, DE ^- y, 



Then FD = yQjO%^, TP = :cCos(7r-^) = — a:Cos^. 



.-. FT^-PD-DF-TP. 



=; d —J) — yCos<f) + xCosO. 



