444 Transactions. — Miscellaneous. 



between them is equal to the length of the seventh bai% 

 CA. 



Elementary geometry shows that the points A, 0, P will 

 always lie in a straight line, and also that BD is at right angles 

 to AP, and bisects it. 



Hence (by Euclid, ii., v.) the rectangle AO OP is equal to 

 the difference between the squares on AE and EO, or to the 

 difference between the squares on AB and BO — that is, the 

 rectangle AO OP has always the same value in w'hatever 

 manner the bars may be tun:ied round their hinges. But as 

 they turn the point A moves on a circle whose centre is C and 

 radius CA, and which therefore passes through O, since CO is 

 equal to CA. Hence P must move on a straight line perpendi- 

 cular to CO. 



If the distance CO be taken different from the length of 

 CA the point P will describe a portion of a circle. 



A second method of producing a rectilinear motion by link- 

 work, depending on the same geometrical proposition, requires 

 only six bars, equal in pairs. 



Four of these are linked together so as to form an anti- 

 parallelogram, ABDC ; AB, CD being equal bars, and also 

 AC, BD. Then it is a consequence of elementary geometry 



that, if P, Q, E be three points on the rods AC, AB, CD re- 

 spectively such that PQR is parallel to AD or CB in any 

 one position of the framework, they will ahvays satisfy this 

 condition in whatever way the bars be turned about their 

 joints. 



The remaining pair of bars are linked to KG and AB at P 

 and Q, and to each other at 0. 



Then the ratio of PQ to BC is fixed throughout the motion, 

 and also that of PR to AD. Hence the ratio of the product 

 of PQ and PR to that of CB and AD is given. But this latter 

 product is invariable (Euclid, vi., D). Hence, also, the former 

 has always the same value. 



Thus, if OP be fixed, since Q must describe a portion of a 

 circle, with O as centre, passing through P, the point R will 

 describe part of a straight line perpendicular to OP. 



