S8 



OF ACCELERATING FORCES. 



ACE -RDV 



G Let A and B be 



" ' ' ' — ■— ' • . 



moving in the same 



line, and while A describes AC, let B describe BD; 

 then while A describes CE=:AC, B will describe DI'"^ 

 BD. For suppose AC=2BD, and let AG=2AB, then 

 AB and BG have been equally decrca<;ed in one instance, 

 and tlic relations remaining the same, they will still be 

 equally decreased (217) ; for the relative motion of A and 

 B is<qual to that of B and G, and any absolute motion be- 

 ing no way determinable, there can be no reason why the 

 one should be otherwise affected than the other ; therefore 

 CE will be twice DP : and a similar proof miglu be given 

 in cases more complicated. 



224. Theorem, If any number of points 

 wove in parallel lines, describing equal spaces 

 in equal times, they are quiescent with re- 

 spect to each other ; and if all the points of 

 a plane move in this manner on another 

 plane, either plane will be in rectilinear mo- 

 tion with respect to the other. 



Let A, B, and C, describe, in a 

 given time, the equal parallel 

 lines AD, BE, CF, then ABnDE, 

 EF=BC, and DF=AC (109), 

 J;' and the points are mutually qui- 

 escent (218, 219). 



225. Definition. If a plane be in rec- 

 tilinear motion with respect to another, and 

 if, besides this general motion of the plane, 

 any point be supposed to have a particular 

 motion in it, it will have two motions with 

 respect to the other plane, one in common 

 with its plane, and the other peculiar to it- 

 self; and the joint eifect of these motions 

 with respect to the other plane, is called the 

 result of the two motions. 



£26. Theorem. The result of two mo- 

 tions with respect to a quiescent space is the 

 diagonal of the parallelogram of which the 

 sides would be described by the separate mo- 

 tions ; and any motion may be considered 

 as the result of any other motions thus com- 

 posing it. 



Let A, B, and C, be three quiescent Y XB C 

 points, and let Z, Y, and X, be three / ^ 

 points in another plane which moves ^i____/ 

 in the direction AZ, BY ; then the ^ "^ 



point A has a rectilinear motion with respect to the plane 

 ZYX ; now while AZ is described by Z, let A have a mo^ 

 tion in its own plane equal to AB ; then it will have two 

 motions with respect to ZYX, by the joint effect of which 

 it will arrive at X in that plane ; and if the motions are 

 both equable, it may be shown by the properties of similar 

 triangles, that it describes the diagonal ZX. Now it is of 

 no consequence to tha relative motion of A and ZXY, 

 which, or whether either, be imagined to be absolutely at 

 rest ; therefore, in general , the result of two motions in a 

 quiescent space, is the diagonal of the parallelogram of 

 which the sides would be described by the separate motions. 

 And the motion thus produced is precisely the same as if 

 derived from a simpler cause. 



227. Theorem. Any equable motions 

 represented by the sides of a triangle or po- 

 lygon, supposed to take place in the same 

 moveable point, in directions parallel to those 

 sides, and in the order of going round the 

 figure, destroy each other, and the point re- 

 mains at rest. 



For two sides of the triangle, AB, BC, JB 



are sides of the parallelogram ABCD, 

 therefore by the motions AB, BC,or AB, A 

 AD, A would arrive at C, while by the 

 motion CA it would be brought back to A J) 

 in the same time ; and all the motions being equable, it 

 will always remain in A. In--the same manner the proof 

 may be extended to any number of sides ; and the truth of 

 the proposition will also appear by considering several suc- 

 cessive planes as moving on each other, and the point A as 

 moving in the last. 



SECT. II. OF ACCELERATING FORCES. 



228. Definition. Any immediate cause 

 of a change of motion is called a force. 



ScHOLruM. The essential nature of force is unknown to 

 us ; even in cases of apparent impulse, the bodies are not 

 actually in contact. When a body is once in motion, it 

 Deeds no foreign power to sustain its velocity (223) ; and the 



