244 Proceedings of the Royal Irish Academy, 
of the circle are in the same straight line. Further, the straight 
lines in which all the points of the circle move pass through the 
same point Q. 
All the points of the circle we have found are in points of inflec- 
tion upon the curves described by each point during the entire 
movement. Two points can however be found upon the circle at 
which the tangent passes through four consecutive points of the 
corresponding curve. 
Let the circle B have rolled till 0' is the point of contact, and 
let LXF(rig. 2) be the new position assumed by the circle OPQ, 
(Pig. \) Q having remained behind. All the points in LXY have 
been moving in lines passing through Q. Let 0" be the new instan- 
taneous centre. Join 0" Q and upon 0"Q as diameter desciibe a 
circle, the points XP'will still continue to move 
on lines through Q. Hence four consecutive po- 
sitions of XY will lie upon the same straight 
line. 
The points which we have determined are 
the points which most nearly describe straight lines. 
It will be seen that this general investigation in- 
cludes what are known in mechanism as the parallel Fig. 2. 
motions. 
n. SLIDING CONTACT. 
/ 
{a) Two cams in a plane, each rotating about a point, are in con- 
tact. It is required to determine, by a geometrical construction, the 
consecutive position of the point of contact, and also the points upon 
the cams which will touch at that point of contact. 
The cams may be replaced by their circles of curvature at the 
point of contact for two consecu- 
tive positions. Let X and I^be 
the centres of curvature, Pig. 3, A 
and B the centres of motion. It 
is known that Z is the instanta- 
neous centre about which the line 
XY turns ; hence the motion of 
the point T is in the line TT' per- 
pendicular to TZ] T is therefore 
the next point of contact of the two 
cams. Ey describing circles T'M 
about A, and T'L about J5, the 
points M and L upon the cams 
which ^vill come into contact at T are found. 
^ {I) Two cams in a plane are in contact. It is required to deter- 
mine a geometrical representation for the circumstances of their mutual 
action. 
