166 
Proceedings of the Royal Society 
Therefore the triangles are equiangular (Euc. VI. 6), wherefore the 
angle qCA = P^qA . But P/qA, the angle in a semicircle, is a 
right angle (Euc. III. 31), therefore qCA is also a right angle. 
With any other position p 2 ofP, C will take up a corresponding 
one c 2 , by joining which with C a right angle is formed. Where- 
fore the point C moves in a straight line at right angles to AB 
produced. 
When the ratio subsisting between the length of the radial bar 
and the distance of the pivot from the fulcrum is not one of 
equality, the position of the parallel point is a circle, which is 
concave or convex with respect to the fulcrum, according as BP is 
greater or less than AB. The results, as determined by methods 
of analysis, show that the equation is the same for both the sym- 
metrical and non-symmetrical forms of the cell ; that is, for both 
the ordinary cell, where the pivot B in the initial position lies in 
a line with the fulcrum and the poles, — and for that form where 
the line passing the poles makes in the initial position a tangent 
to the circle described by the radial bar. Thus, if R were the centre 
of the circle described by C, its distance from the fulcrum A, 
is RA = 
AB [AM 2 - MP 2 ] 
BP 3 - AB 2 ; 
and the length of the radius 
BP [AM 2 - MP 2 ] 
IS iiu - B p 2 _ AB 2 
The determination of the position of R, which is evident in the 
non-symmetrical form of the cell, is simply given in the ratio 
RC : RA : : BP : AB. 
Thus, if AM = 7; MP = 5; BP = 3; andAB = l 
th „ BP [AM 2 - MP 2 ] 3(49 - 25) 
tuen JUj - Bp2 _ AB 2 - 9 _ j =9 
RA = ~ • RC = 3 
One form of the cell has given rise to a discussion of the question 
whether the parallel motion of Peaucellier is not simply a modifi- 
cation of the pantagraph. The resemblance between the two 
systems is not noticeable at first sight ; and one would be inclined 
to deny any connection between them. 
