472 



MOTION. 



running are fulfilled if the vital force which 

 the body receives hy the extension of the 

 supported leg whilst it is on the ground, is 

 equal to the vital force communicated to 

 the swinging leg during the time of its oscil- 

 lation.* 



The motions of walking and running tend 

 continually to approximate, in proportion as the 

 time in which both legs are on the ground in 

 the former, and the time in which the body is not 

 supported at all in the latter, is diminished, and 

 the laws of both running and walking coincide, 

 when the time that the body swings unsup- 

 ported in the air in the former, and that where 

 both legs are on the ground in the latter, 

 vanishes. 



The resistance of the air to the trunk, when 

 it is propelled with great velocity as in run- 

 ning, requires to be compensated, in order to 

 be kept in a state of equilibrium, either by 

 the force of its own gravity, or by that of its 

 muscular system. The former method is usually 

 adopted to prevent an unnecessary expenditure 

 of vital power. 



It will be seen in Table 4, that the angle of 

 inclination of the trunk in running, is to that in 

 quick walking, as 49.8 to 31.8, when the 



~ 



(l + rii*5) 8 (1 



... (33) 



(34) 



(35) 



In which equations, 



T 1 



8 = arc (cos \ 



n= l + 



lgt* 



The other expressions being the same as those in 



walking, 

 c .... is the uniform velocity of the point m in 



a horizontal direction. 

 s .... is the space through which the centre of 



gravity is raised in the time t, 

 '.... is the space through which the same 



centre is raised in the time t (J. 

 When T t o, that is, in slowest running, 



the above equations become identical with those 



in quickest walking, where the time during 



which both legs are on the ground vanishes. 



* That is, when 



m' (I + r n t -)i c- m ' (I r) 2 c 3 



and as /* _ ( by substituting p, we get equa- 

 tion (33), from which the height of the centre 

 of gravity is to be found. 



length of step in the former is to that in the 

 latter as 1 .509 to 0.888. 



In order to find the amount of the vertical 

 undulations of the trunk in running, MM. 

 Weber viewed the runner through a telescope, 

 adapted for that purpose. They calculate the 

 undulations to be from fth to fth of an inch ; 

 they estimate the duration of the step to be 

 Jth to ith of a second, of which the body 

 swings freely in the air -^th of a second, and 

 of this falls -j'jth ; now, by the law of falling 

 bodies, the centre of gravity will descend in this 

 time 22 mm. = 0-85614 in. ; hence, the body 

 falls through a less space in running than in 

 walking. 



The simultaneous positions of the centre 

 of gravity and of the two feet at individual 

 instants of time, of which ^/fg. 262 is a vertical, 

 and Jig. 263 a horizontal representation. The 

 right foot is marked , the left 6, and the centre 

 of gravity c : 6 t 2 signifies that during the 

 time is passing from to 2 , and c from 

 c to r 2 , b remains standing on the point 6, 2 , 

 and so on. In order to distinguish the swing- 

 ing from the supporting leg, in jig- 262, the 

 former is so contracted that it does not reach 

 the horizontal line ; and in Jig. 263, the swing- 

 ing leg is represented as if it swerved from the 

 path both on the right and left sides alter- 

 nately : r,, r 3 = c y c 5 = r s , r 7 , indicate the 

 length of the step = ju, a v 3 4 = 3 4 > "7 s 

 = b l 2 , 6 5 6 = 2 p. The time which elapses 

 from the instant when 6 steps in 6, 2 to a, in 

 a 3 4 , is the duration of a step T. The time in 

 which c advances from c, to c 2 is the time t, 

 when the body is supported upon one leg. 

 The time in which c advances from r 2 to r a is 

 the time r t, during which the body swings 

 in the air. The time in which c advances from 

 r 2 to r s , is the time in which the left leg 

 swings, which is greater than T, in which c 

 merely passes from c 3 to r s . The instant when 

 the left leg steps in 6, or the right in 3 , the 

 leg must press against the ground with such 

 force as to impede the accelerating force of 

 gravity upon the body, and communicate to it 

 an ascending movement; to accomplish this, 

 the leg must be set down on the ground per- 

 pendicularly, therefore the lines c,, b t , c 3 , 3 , 

 &c. are vertical. 



The length of the extended leg being 6 2 c 2 , 



Jig. 262, at the end of the time t, c , d, = t> 2 d 



~ c t,* the horizontal distance of the body in 



the time t, and r 2 d ~ r, 6 2 + c 2 d l ~ 



h + s, where /< represents the distance of the 



centre of gravity from the beginning of the 



step, and s the height to which it is elevated 



in the time t ; now, if 6 2 d r 2 be a right-angled 



triangle, then 



(b, r,)' - (b a d? + (r a d? 



or, 

 ^ c 5 t* + (/i + s)* 



That is, the square of the length of the extended 

 leg is equal to the sum of the squares of the 



* The line c 2 d is vertical, and c, d t horizontal, 

 meeting c a d in d l ; the letter d and line c, d, are 

 omitted in the figure to prevent confusion. 



