TRANSAGLIONS OF RHE SECTIONS. 17 
On some Conversions of Motion*. 
By H. Harr, M.4A., late Fellow of Trinity College, Cambridge. 
1. The positive and negative Peaucellier cells may be combined together by the 
addition of two extra bars to either, thus making a complete double cell composed 
of two rhombs jointed together, giving four points, A, a, 8, B, lying always in a 
straight line—a, 8 being the poles, A or B the fulcrum of a positive, and A, B being 
the poles, a or 8 the fulerum of a negative Peaucellier cell. Thus each cell has 
two fulcra as well as two poles, the convenience of the second fulcrum being mani- 
fest in the tracing of certain discontinuous curves. As an example, let A, B be 
the poles, a, 8 the fulcra of a negative cell, A’, B', a’, 8’ the poles and fulcra of an 
exactly similar cell, c* being the modulus of either. 
Let A, B' be fixed and A’, B be connected by a bar whose length 26=AB’. Let 
the fulera B, B' be fixed together. Then if c be > 6, 8 will trace out the continuous 
oval of Cassini, but if c be <b, B will trace out an oval surrounding B; if B, B’ be 
set free, and a, a’ fixed, this point a will trace out the conjugate round A. A similar 
statement may be made with regard to the positive cell. 
N.B. While § traces the Cassinian, a and a’ trace out a certain Cartesian ft. 
2. If four bars, AD, BC, AB, CD, equal two and two, be jointed so as to form the 
equal diagonals and equal but not parallel sides of a trapezium ABDC, and if 
QPP'Q' be a straight line parallel to AC or BD, cutting the four bars above in the 
points P, P’, Q, Q’ respectively, then these points are such that 
QP, QP’=Q'PP, Q'P'=PQ, PQ’=P’Q, P'Q'=constant ; 
hence Q, P, P’, Q' are equivalent to the four points A, a, 8, B of the completed 
Peaucellier cell described above. By means of the above system of four bars it is 
evident that, by the addition of a bar which causes FP or P’ to move in a circle 
passing through Q or Q’, we make P’ or P move in a straight line. This contradicts 
the statement made (I believe) by Professor Tchebicheff, that seven bars at least are 
requisite for the conversion of circular into rectilinear motion. 
3. If, further, one of the four bars, say AD, be fixed, then the point on the bar 
which is equal to the one fixed describes the inverse of a conic; for if AD=BC= 
2a, AB=CD=28, and c=the distance of the fixed point P from the centre of the 
bar AD, it is easily seen that the equation to the locus of P’ is 
4b? — 4a? sin?6= { p —2c cos 6}, 
which, when inverted and transformed to Cartesian coordinates, becomes 
(0° —c*)x* +-(b° — a’) y? 4+2ck?a=x' (2x? being the modulus). 
4. Professor Sylvester (to whom I am indebted for various suggestions and 
remarks connected with this paper) has shown that three points otherwise free may 
be made collinear by the use of fifteen bars; by help of the above trapezium motion 
n points may be made collinear by the use of 5 —4 bars. 
ie AOA’ be three points always collinear, and AA’ be moreover the angular points 
of a rhomb APAP’, whose sides are of any constant length, then OP’ must always 
=OP. Hence any number of points otherwise free may be caused always to lie on 
a sphere of variable radius: 
5. Let A, B, C be three points constrained to move along three axes, Ox, Oy, Oz ; 
. let the points A, B be connected by a pantigraph so that a point E always bisects 
AB; let D be connected with O by another pantigraph, so that the middle point 
of OD coincides with E: then OADB is always a parallelogram in the plane «Oy. 
In a similar manner let a parallelogram CODP be constructed ; then OA, OB, OC 
are the coordinates of P. 
Let another series of pantigraphs determine a point P’ from OA’, OB’, OC’, and 
let A, A’, &c. be connected in such a manner that 
OA=f(0A'), OB=$(OB'), OC=¥(0C’), 
* For the paper iz extenso, see ‘Messenger of Mathematics,’ No. 42, New Series. 
1 Its equation is of the form 7,=yp, 7 and p being vectorial coordinates. 
