276 APPLIED MECHANICS 7 
the force required to produce the sliding of the piece mounted on it. A 
convenient way of applying this principle to the guiding of a sliding 
piece so as to reduce the force required to slide it is shown in Fig. 421, 
A A? Fl ae a 
MI S. iD 6 a x1 
rae 4 
De ipt ia 
poy te Se 
oo 
Fi. 422, 
where A and B are two parallel shafts or spindles, which are rotated, 
preferably in opposite directions, and which support the piece C, which is 
made to travel along the shafts by a force T, 
The theory of the action of the rotating guides is as follows. Let 
AB (Fig. 422) represent a horizontal flat plate, upon which rests a body 
D. In a given time, let AB travel a distance AA’ in the direction OX 
into the position A’B’.” In the same time, let the body D be made to 
slide on AB a distance CN in the direction OY at right angles to 
OX into the position D’. The motion of D relative to AB will be the 
same as if while it slides the distance CN in the direction OY it be made 
to slide the distance CM equal to A’A in the direction XO. These 
simultaneous motions given to D will result in a motion of D relative to 
AB in the direction CL, and equal to CL where CL is the diagonal of 
the rectangle MN. Now the force P, acting along CL, necessary to slide 
D along CL, is equal to »R where R is the force, normal to AB, and 
pressing D on AB. But the force P, represented to scale by CL, may be 
replaced by the forces Q and § represented to the same scale by CN and 
OM respectively, and the ratio of the force Q to the force § is evidently 
the same as the ratio of the velocity of D in the direction OY to the 
velocity of AB in the direction OX. Applying this to a rotating guide, 
* a force equal and opposite to S is the tangential force at the surface of 
the guide in the direction of its motion necessary to drive it, and Q is the 
force on the sliding piece in the direction of its motion necessary to make 
it slide. : 
To prevent D being carried in the direction OX when AB moves 
under it in that direction a fixed guide EF is necessary, and the force ~ 
pressing D against this guide is evidently equal to 8, which will cause a ‘ 
resistance equal to »S in the direction YO. Hence the resultant force 
necessary on D in the direction OY is Q+pS. By using two rotating 
guides rotating in opposite directions, the tractive force on the sliding 
piece is reduced from Q +S to Q for each guide. 
To prevent AB moving in the direction OY when D moves over itin 
that direction a fixed guide HK is necessary, and the force pressing AB 
against this guide is evidently equal to Q, which will cause a resistance 
equal to »Q in the direction XO. Hence the resultant force necessary 
on AB in the direction OX is equal toS+pQ. Ina rotating guide the 
resistance which would correspond to the resistance »Q would be the 
