18 LECTURE II. 



In all these cases it is of importance to attend to the composition 

 of motion, or the joint effect of more than one motion existing at the 

 same time. The existence of two or more motions at the same time, 

 in the same body, is not at first comprehended without some difficulty. 

 It is in fact only a combination or separation of relations that is con- 

 sidered ; in the same manner as by combining the relation of son to 

 father, and brother to brother, we obtain the relation of nephew to uncle, 

 so, by combining the motion of a man walking in a ship, with the motion of 

 the ship, we determine the relative velocity of the man with respect to the 

 earth's surface. It is, however, necessary, for ascertaining these relations, 

 to consider the affections of a space or surface in motion, and to examine 

 how it may move in the 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 rectilinear distance of each, from each of 

 the rest, remains 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 respect to the other. This is easily exemplified 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 [car- 

 penter's] square is made to slide along the straight edge of a board, the 

 surface of the square is in rectilinear motion with respect to the board. 

 (Plate I. Fig. 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 re- 

 spect to the other plane : the one in common with its plane, and the other 

 peculiar to itself ; and the joint effect of these motions, with respect to the 

 second plane, is called the result of the two motions. Thus, when a car- 

 riage moves on a perfectly [straight and] 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 which is 

 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 what- 

 ever position the arm be fixed. Here we have two motions in the slider, 

 one in common with the arm, and the other peculiar to itself, which may 

 be either equal or unequal to the first ; and by tracing a line on a fixed 

 plane, with a point attached to the slider, we may easily examine the joint 

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



The joint result of any two motions is the diagonal of the parallelogram 

 of which the sides would be described, in the same time, by the separate mo- 



