472 
running are fulfilled if the vital force which 
the body receives by 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- 
eg in the air in the former, and that where 
th 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 
2s 
(4 +1) 8 (¢— 6" (1— 55) 
soe (33 
(ratty —(—ry GC 
ea eee eee weee 
<<" 
vise (96) 
(35) 
*eeeeeere 
In which equations, 
ee, 
Le = are (cos 
r 
rn= Vi) <b r(h — s)" 
lg t 
The other expressions being the same as those in 
walking, 
¢ .-.. isthe uniform velocity of the point m in 
a horizontal direction. 
$ se++ is the space through which the centre of 
gravity is raised in the time ¢. 
s'..+- 1s the space through which the same 
centre is raised in the time ¢ — 6. 
When + —¢ = 0, 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 
1 SE (¢— OY 2a _ 
(m + m’) rane ene ¢| 77) = 
m (l-brat yc —m(1— ry ct 
and as == by substituting «, we get equa- 
tion (33), from which the height of the centre 
of gravity is to be found. 
} 
MOTION. 
length of step in the former is to that in the 
latter as 1.509 to 0.888. ~ ete 
In order to find the amount of the 
undulations of the trunk in running, I. 
Weber viewed the runner through a tele cope, 
adapted for that purpose. They calculate the 
undulations to be from 3th to §th of an inch; 
they estimate the duration of the step to be 
jth to jth of a second, of which the bod) 
swings freely in the air jth of a second, and 
of this falls jth; now, by the law of falling 
bodies, the centre of gravity will descend in this 
time 22 mm. = 0°85614 in.; hence, the dod) 
falls through a less space in running than in 
walking. ‘nd 
The simultaneous positions of the centre 
of gravity and of the two feet atindividua 
instants of time, of which fig. 262 is a vertic 
and fig. 263 a horizontal representation. The 
right foot is marked a, the left b, and the centre 
of gravity c: 6,, signifies that during th 
time a is passing from atoa,, and ¢ fro 
c to ¢,, b remains standing on the point 6, , 
and soon. In order to sey the swing- 
ing from the supporting leg, in fig. 262, th 
Srnler is so sa that it does not reac 
the horizontal line ; and in fig. 263, the swin 
ing leg is represented as if it swerved from th 
path both on the right and left sides alter 
nately: ¢,, Cy = C3, Cs = Cy, Cz, indicate the 
length of the step =p, a, dy, = Gsy 
= 6b, » 6,6 = 2p. e time which elaps 
from the instant when 6 steps in 6, , toa, i 
a, 4, is the duration of astep vr. The time ii 
which ¢ epee vey frém c, to c, is the time 
when the body is suppates upon one le 
The time in which c adeincan fiom Cy 0 ¢, 
the time + — ¢, during which the body s' Dy 
in the air. The time in which ¢ advan ro 
c, to c,, is the time in which the left k 
swings, which is greater than +, in which 
merely sg from c, toc,. The instant wher 
the left leg steps in 6, or the night in a,, t 
leg must press against the ground with su 
force as to impede the accelerating force — 
gravity upon the body, and communicate to 
an ascending movement; to accomplish th 
the leg must be set down on the ‘?p 
pendicularly, therefore the lines ¢,, 6, €gy: 
&c. are vertical. a 
The length of the extended leg being 6, 
Jig. 262, at the end of the time ¢, ¢, d, = 
= c t,* the horizontal distance of the body 
the time ¢, and c, d = ¢, b, +o, a 
h +s, where / represents the distance of 
centre of gravity from the beginning of 
step, and s the height to which it is eley 
in the time ¢; now, if 6, dc, be a right-am 
triangle, then a 
(b, 2)? = (6, d)? + (c, dP . 
or, 
RP=COP +E AESP 
That is, the square of the length of the exter 
leg is equal to the sum of the squares of 
* The line cyd is vertical, and ¢, d, horia 
meeting c,d ind, ; the letter d and line ¢ 
omitted in the figure to prevent confusion. — 
3 
-, 
