28 



MECHANICS. 



Let / be any load less than L, and let 

 x be the greatest speed with which this 

 load can be carried. The useful effect 

 will be / x x ; that is, the load multi- 

 plied by the speed. The rule which 

 seems best to agree with experience is 

 that the load /increases in the same 

 ratio as the square of the difference 

 between the greatest velocity X, with 

 which the animal can move unloaded, 

 and the greatest velocity x with which 

 it can move the load ; that is, / increases 

 as (X .r) 2 . Assuming this rule, 

 therefore, it follows, that the useful 

 effect is represented by the product 

 (X x) 2 xx. This will probably be 

 more easily understood by reducing it 

 to an arithmetical table. Let us sup- 

 pose that the number 15 represents the 

 greatest unloaded speed, and that the 

 Speed - - - 1 2 3 4 5 

 Load .... 225 196 169 144 121 100 



square of 15, or 225, represents the 

 greatest load which can be sustained 

 without moving. The signification of 

 the units which compose the num- 

 ber 15, will be found, by dividing the 

 space through which the animal would 

 move in a given time, suppose one 

 hour, into 1 5 equal parts : each of these 

 parts will be expressed by an unit of 

 the number 15, which expresses the 

 greatest unloaded speed ; and the signi- 

 fication of the units of 225 will be found, 

 by dividing the greatest load which can 

 be sustained without moving, into 225 

 equal parts : one of these parts will be 

 expressed by an unit of the number 

 225, which expresses the greatest load. 

 The following Table gives for each 

 degree of speed from I to 15, the cor- 

 responding load and useful effect, 



6 7 8 9 10 11 12 13 14 15 

 81 64 49 36 25 16 9 410 



Useful effect 196 338 432 484 500 486 448 392 324 250 176 108 52 14 



From the inspection of this Table it 

 appears that a much greater useful 

 effect is to be attained by the slower 

 motions with heavier loads than by the 

 quicker motions with lighter loads. The 

 greatest useful effect is produced by the 

 speed 5 with the load 100 ; that is, with 

 a velocity which is one- third of the 

 greatest unloaded speed, and a load 

 which is four-ninths of the greatest 

 load which can be sustained without 

 moving. \Ve shall find this result, 

 whatever be the number we take, to 

 represent the greatest speed. * 



* The mathematical investigation is not difficult. 

 Let M be the useful iffect, Then by the empirical 

 formula already explained we have M =; (X x^x. 

 pifferentating this we obtain 



Supposing this =0 we shall obtain the value of x, 

 which corresponds to a maximum or minimum value 

 of v. This gives the equation, 



(X-.r) (X-3.r) =0 

 the roots of which are 



V 1 Y 



a: = X ,r = X. 



For T=X the load and useful effect are each =0. 

 This root, therefore, corresponds to a minimum ; and 

 for x = X f:=(X -J-X)2 =1_X2 ; that is, 



3 8 ' 9 



the load corresponding to one-third of the greatest 

 speed is - of the greatest load ; for L = X 2 . That 



this is a maximum is easily shown by taking the 

 econd differential, which gives 



-- = - 3 (X - .r) - (X - 3*) 



- _ 4 X 4 G.r 

 which, if we substitute- X for .r, we find 



Which, being negative, shows that the value X 

 Cprresponds to a maximum value of . 



Thus, if the greatest unloaded speed 

 of a horse be 1 5 miles an hour, and 

 that the greatest weight which he is 

 capable of sustaining without moving 

 be divided into 225 equal parts, his 

 labour will be most advantageously 

 employed if he be loaded with 100 of 

 these parts, and travels at the rate of 

 5 miles per hour. If he be thus em- 

 ployed, it will be found that he will 

 carry a greater weight through a given 

 distance in a given time than under any 

 other circumstances. 



The average value of human strength, 

 considered as a mechanical agent, has 

 been variously estimated. Desaguliers 

 considers that a man can raise the weight 

 of 550 Ibs. 10 feet high in a minute, and 

 continue to do so for 6 hours. Smeaton 

 considers that this is too high an ave- 

 rage, and thinks that six good English 

 labourers will be required to raise 

 21,141 solid feet of sea-water to the 

 height of four feet in four hours. In 

 this case they will raise very little more 

 than six cubic feet of fresh water each, 

 1 feet high in a minute. The labour- 

 ers whom Smeaton supposes capable of 

 executing this work he considers to be 

 equal to twice the number of ordinary 

 men. It would, therefore, perhaps, be 

 a fair average value of a man's work 

 to estimate it, for a continuance, at half 

 an hogshead of water raised through 

 1 feet in a minute. 



The efforts of men differ with the 

 manner in which these efforts are em^ 

 ployed. It has been shown by Mr. R. 



