24 JLECTURE H, 



motion, and to examine how it may mov£ in tlie most simple manner with 

 respect to another space. 



If any number of points move in parallel lines, describing equal spaces in 

 equal times, they are at rest with respect to each other; for it may easily be 

 demonstrated that the rectiUnear distance of each, from each of the rest, re- 

 mains unchanged: and if all the points of a plane move in this manner on 

 another plane, either plane may be said to be in rectilinear motion with re- 

 spect to the other. This is easily exemphfied by causing one plane to move 

 on another, so that two or more of its points shall always remain in a given 

 right line in the second plane : as when a square is made to slide along the 

 straight edge' of a board, the surface of the square is in rectilinear motion 

 Avith respect to the board. (Plate L ¥ig. 4.) 



If, besides this general motion of the plane, any point be supposed to have a 

 particular motion in it, the point will have two motions with respect to the 

 other plane, the one in common with its plane, and the other peculiar to it- 

 self; and the joint effect of these motions with respect to the second plane is 

 called the result of the two motions. Thus, when a carriage moves on a per- 

 fectly level road, all its points describe parallel lines, and it is in rectilinear 

 motion with respect to the road: its wheels partake of this motion, but have 

 also a rotatory motion of their own ; and the result of the two motions of each 

 point of the wheels is the cycloid or trochoid that it describes in a quiescent 

 vertical plane. (Plate I. Fig. 5.) 



When an arm is made to slide upon a bar, and a thread, fixed to the bar, 

 is made to pass, over a pulley at the end of the arm next the bar, to a slider 

 which is moveable along the arm, the slider moves on the arm with the same 

 velocity as the arm on the bar; but if the thread, instead of being fixed to the 

 slider, be passed again over a pulley attached to it, and then brought back to 

 be fixed to the arm, the motion of the slider will be only half that of the arm ; 

 and this will be true in whatever position the arm be fixed. Here we have 

 two motions in the slider, one in common with the arm, and the other pecu- 

 liar to itself, which may be either equal or unequal to the first; and by trac- 

 ing a line on a fixed plane, with a point attached to the slide/, we may easily 

 examine the joint result of both the motions. (Plate I. Fig. 6.) 



