462 



MOTION. 



move uniformly in a horizontal line, it has a 

 retrograde motion in its curve of oscillation, 

 which, if measured from its lower extremity, is 

 just equal to the velocity of the point of junc- 

 tion, when estimated in the direction of the 

 tangent of the curve at the commencement of 

 the step. Hence, in quick walking, when the 

 swinging leg is supposed to come to the ground 

 in a vertical position, it describes half of the 

 curve during each step. By following the 

 same course of reasoning Messrs. Weber have 

 ascertained the amount of the vis viva, or vital 

 force, communicated to the swinging leg in 

 any given time ; the amount communicated to 

 the body by the standing leg; the proportion 

 which the time when the body is supported on 

 one leg bears to the whole time of a step ; and 

 the height at which the centre of gravity is 

 borne above the ground.* In order to verify 



* After an elaborate analysis the Messrs. We- 

 ber have deduced the following equations, which 

 express the general laws of walking. 



(23) 

 (24) 



IT 



- = 1 + n Cos \ JL (t 



:= k (r Q 



.. (25) 

 .. (26) 



In which equations 



r/i 1 



JL arc (Cos 



it 



*>. 



The mass of the trunk is supposed to be con- 

 centrated in a point m at the upper end of the 

 le-, and the mass of the swinging leg in a point 

 H'in the leg which is considered as a straight 



line. 



/ is the length of the hinder leg at the begin- 

 ning of a step. 



h is the height of m above the horizontal plane 



at that time. 



1) is the distance between the hinder toot and 

 the forward at that time, or the length of a step. 



6 is the time during which m falls below its ho- 

 rizontal line at the end of the time t. 



f is the length of the hinder leg at the end of 

 the time 3 before it is extended or becomes I. 



r is the time of one step. 



t is that portion of it during which the leg is 

 swinging. 



the pendulous movements of the legs a person 

 was placed upon a small block, and by sup- 

 porting himself on one leg, suffered the other, 

 measuring thirty inches, to swing, with relaxed 

 muscles, as a pendulum. A vibratory motion 

 havingbeen communicated by a slight movement 

 of the trunk backwards and forwards, the num- 

 ber of oscillations made in the time of a minute 

 were found to be 84, consequently 



fin" 



_ = 0".714285 = time of one oscillation ; 



84 



and since the lengths of pendulums at the 

 same place vary inversely as the squares of the 

 numbers of oscillations in a given time, 

 84* : 60* : : 39| : I (the length of a pendulum 

 which vibrates synchronously with the leg) ; 



, 60* X 39A 140850 



hence 1= _ ? = = 19.961 



84 2 84 2 



inches. Now as the whole length of the leg 

 was 34 inches, the centre of oscillation must be 

 less than two-thirds of that length from the 

 point of suspension, and consequently less 

 than in a prismatic rod, the length of which 

 is such as will vibrate synchronously with 

 the leg. This accords with the known figure 

 of the leg, the mass of which diminishes as 

 the distance from the axis of motion in- 

 creases. The time of a half oscillation of 

 the freely suspended leg which we have found 

 .714285 _ ".357i425, approximates very 



closely to that found by the Webers, both in 

 the living and the dead subject. 



The second experiment was made by our 

 engraver, Mr. Vasey. In walking at the rate 

 of four miles per hour he counted 2000 steps 



every fifteen minutes; then 





2000 



6 = 0".45 



the time of each step ; now as 2000 steps were 

 taken in one mile of 5280 feet, the length of each 



r is the ratio of the distance between m and 

 m' to /. 



g is the accelerating force of gravity = 32.5 

 feet. 



it is the number 3.1416. 



T is the time of an oscillation of a pendulum 

 whose length is r/. 



fj. is the ratio of the mass 'of the trunk to the 

 mass of the swinging leg. 



The quantities i T, p. must be previously 

 ascertained, and it and g being always the same, 

 there will be nine equations for finding the values 

 of the ten remaining quantities ; if therefore we 

 know any one of these ten, the rest may be found. 



In regular progression the force communicated 

 by the supporting leg to the trunk must equal 

 that imparted by the trunk to the swinging leg, 

 that is 



from which, by substituting L for c and (*. for 



we get equation (26.) 

 m 



