408 



MOTION. 



ar swimming, flying, crawling, climbing, 

 leaping, running, walking, &c. The con- 

 sideration of these diversified methods of 

 progression involves the theory of the mo- 

 tion of bodies in general, of the lever, the 

 pulley, the centre of gravity, specific gravity, 

 and the resistance of fluids, &c. ; and, as 

 we shall have occasion for constant refer- 

 ence to the mechanical principles connected 

 with these subjects, they will be first dis- 

 cussed; but for the convenience of those 

 who are unacquainted with the algebraic 

 method of computation and analysis, the 

 latter will generally be separated from the 

 text. 



Fundamental Axioms. First, every body 

 continues in a state of rest or of uniform mo- 

 tion in a right line until a change is effected by 

 the agency of some mechanical force. Secondly, 

 any change effected in the quiescence or mo- 

 tion of a body is in the direction of the force 

 impressed, and is proportional to it in quantity. 

 Thirdly, reaction is always equal and con- 

 trary to action, or the mutual actions of two 

 bodies upon each other are always equal and 

 in opposite directions. 



Thus if M (Jig. 203) be a particle of matter 

 free to move in any direction, and if the lines 

 MA, MB, represent the intensity of two forces 



Fig. 203. 



A 



-B 



acting on it in the direction MC, the particle M 

 will move towards C by the combined action 

 of the two forces, and it will require a force 

 in the direction of CM, equal to MA+ 

 MB to keep it in a state of rest: but if MA 

 and MB (Jig. 204) represent the intensities 

 and directions of two forces acting on the par- 

 ticle M in opposite directions, if MA be 



Fig. 204. 



M 



greater than MB, the particle M will be 

 moved towards A by the difference of these 

 two forces, and it will require a force equal to 

 that difference to keep it at rest. 



The composition and resolution of forces. 

 In the composition of forces it is proposed to 

 find the resultant, arising from any number or 

 system of forces acting upon a given point. The 

 resolution of forces, which is the converse of 

 the former process, consists in discovering what 

 forces acting in given directions would com- 

 bine to produce a given resultant: Thus, if there 

 be two forces F F' (Jig. 205), whose directions 

 and magnitudes are represented by F N, F' N, 

 and if FR, Fit be drawn respectively parallel 

 to F'N, FN, then by the composition of forces 

 we find the magnitude and direction of the 

 equivalent or resultant of these two forces to 

 be RN, and conversely it may be resolved 

 into a pair of forces as RF, RF' represented by 

 the adjacent sides of any parallelogram, of 



which RN is the diagonal, 

 and consequently into an 

 indefinite number of such 

 pairs.* This construction is 

 called the parallelogram of 

 forces. 



The resultant of any 

 number of forces meeting 

 in a common point may be 

 ascertained thus : first, let 

 the resultant of any two 

 forces be found as before, 

 and substitute this one force 

 for the two components pro- 

 ducing it ; then combine 

 this new force with one of 

 the remaining forces, and continue this process 

 until all the iorces are reduced to a single force, 

 which is the resultant sought. The following 

 geometrical solution will render the subject 

 more apparent : let P, P', P", &c. (jig. 206) 

 represent a number of forces meeting in the 

 common point A, and let A P, A P', A P" be 

 proportional to these forces respectively : 

 through P draw PR equal and parallel to 

 AP', and through R draw RR' equal and 

 parallel toAP", and through R'drawR'R" 

 equal and parallel to A P"'; join A R", which 

 represents the resultant of the four forces A P, 

 AP', AP", AP'". A similar opera- 

 tion will serve for any number of forces. 

 This figure is denominated the poly- 

 ......... C gon of forces. If the directions of 



three forces are rectangular, and in dif- 



ferent planes, the resultant may be found 



as follows: let PC, PC', PC", (fig. 207) be 



the intensities and directions of three forces, 



complete the parallelepiped BD; then the 



forces PC, PC' have P r for an equivalent, 



therefore P r may be substituted for these two 



forces; and by compounding the forces PC", 



P r, we get PR the diagonal of the parallele- 



piped BD r for an equivalent to the three 



forces. This construction is called the 



parallelopipedon of forces.^ 



An equilibrium cannot subsist be~- 



- iB tween any two forces acting upon a 



point of matter, if the lines represent- 



ing the directions of the forces be inclined 



to each other at any angle ; but a third force, 



equivalent to their resultant and in an oppo- 



Vide Gregory, vol. iii. ch. ii. p. 23. 

 t From the same construction the resultant of a 

 system of forces maybe found, disposed in different 

 planes by the method of rectangular co-ordinates : 

 let P C = T, P C' = y, P C" = z, (fig. 207), 

 by the 47 Euc. Lib. I. we have Pr 2 '= PC'4- 

 2 , and PR 2 =Pr 2 + rR 2 = 

 + y' 2 + z 2 , whence the resultant PR = 



s/ .r 2 + (/ '-+ z 2 . The position of the resultant 

 is thus determined ; let r, r', r" denote the un- 

 known angles formed by the direction of the re- 

 sultant with each of the co-ordinates, and R cos. 

 r, R cos. r', R cos. r" will represent the equivalents 

 of the resultant in the several directions ot the axes, 

 hence we have R cos. r = x, R cos. r' ^ (/, 

 ii cos r" = z, .-. 



Cos. r = -5' cos. r' 



it 



y 



. 



cos - r =5' 



