Sept. 26. 1872] 



NATURE 



431 



some other force than that of gravity clue to the assumed gradient. 

 But it may be shown by the o:her method of consideiirg the 

 problem, by assuming a motion tuwanls the pole of one mile per 

 day, that the forces whicii overctnie the resistances in both di- 

 rections are about of the same order. This very great difference 

 in the results obtained from the two methods of considering the 

 problem, indicates that there is some great fallacy somewhere 

 which needs looking after. 



Mr. Croll is misled by adopting the erroneous principle that 

 the amount of work performed by gravity upon a falling or de- 

 scending mass is in all cases expressed by the mass multiplied 

 into the height through which it descends, or that the foot-pound 

 is a unit of work. The amount of w ork required to give velocity 

 to any body, or overcome any kind of frictional resistance, is 

 expressed by the intensity of the effort, regarded as constant, 

 multiplied into the time of action. The intensity of the effort of 

 any force, as of gravity, is explained by the mass multiplied into 

 the velocity which such force can produce in a unit of time. If 

 we, therefore, put g, ;«, /, ar.d v fer the force, the mas?, the 

 time, and the velocity respectively, we shall have, putting //'for 

 the amount of work performed by gravity, 



(1) W = mgt - mv, 



that is, the amount of work performed is expressed by the mo- 

 mentum. Now this amount of work is stored away in the mov- 

 ing body, and remains until it is used in overcoming resistance 

 of some kind, as friction or the inertia of other bodies, and IV is 

 exactly the expression of the working power which has been 

 communicated to it. But if the moving body has been subject 

 to resistances, as of friction, during the time /, then we shall 

 have 



(2) /K= mv +ft 



putting / for the coefhcient of friction, and supposing it to be 

 constant. In this case mi' expresses the amount of work left 

 which has not been expended in overcoming the friction during 

 tlie time /, and of course in this case we cannot get all the work 

 back again which has been expended, at least mechanically. 



If we now suppose a body to foil in a vacuum through the 

 space s, if the amount of work performed by gravity upon it, 

 and the working power communicated to it, is expressed by the 

 mass multiplied into the height through which it has fallen, we 

 shall also have 



(3) IV = ms = \'i'gt'^ = \mvt. 



Hence, comparing the preceding expression? of //', we have 

 niv = )jiir<'t, which is impossible ; and therefore, if the former 

 expression of IV is correct, the latter is not. 



Again, illustrating by a special case, if we suppose a body, 

 of which the momentum is inv, to move upon a level plane 

 without friction, and the plane to curve up in the direction of 

 motion, and also suppose another body with half the mass and 

 double the velocity, of which the momentum is \m X 2:/ = mv, 

 m and <' being the mass and velocity of the first body, it must 

 be admitted th.at the amount of labour expended in giving both 

 bodies ihe momentum inv is exactly the same, in the latter case 

 the intensity of the effort being half as great, but the time of 

 action twice as long ; but the momentum will carry the latter up 

 the slope to a height lour times greater than that of the former, 

 and after descending again to the plane, both will have the same 

 momentum, and the same amount of labour would be required 

 to bring each to rest, and consequently both have the same 

 power of doing work. But the mass of the latter multiplied 

 into the height through which it has descended is double that of 

 the former, and hence these products do not express the power 

 of doing work which gravity tias communicated to them. 



In the case in which the descending body is resisted by friction, 

 we have seen (2) that neither the m.ass multiplied into the l.eijht 

 of descent, nor the momentum mv which the body has on arriv- 

 ing at the level plane, expresses the whole action of gravity, and 

 the resistance yi' may be so small as not to affect teniibly the 

 amount of work expressed by i/rv, or it may be so great that 

 mv may be neglected, in comparison with ft. The value oi ft 

 may also be so great that the value which mv would have (i) in 

 the case of a free body falling through the space .r, would be al- 

 most infinitely small in comparison with the whole expression of 

 IV (2). In the various cases, therefore, which may be supposed, 

 in which friction may be either very small or very great, so that 

 in the former case the effect of the reds'ance might be scarcely 

 sensible, and in the latter it might take the action of gravity a 

 long time to drag the body down through the space which it has 

 to descend, we cannot suppose that in all these cases the whole 



action of gravity is expressed by the same number of foot-pounds, 

 supposed to be units of work. 



If, therefore, Mr. Croll's pound of water were moved from the 

 equator to the parallel of 60° by the action of gravity without 

 any resistance, the momentum which it would have on arriving 

 there would express the work done by gravity upon it, and not 

 the si.x foot-pounds, and the work would be done in a very 

 short lime in comparison with the time in the real case of nature ; 

 but when it is dragged down there through all the resistance 

 which it suffers, at the rate of a mile per day, as we have sup- 

 posed, the amount of labour which gravity performs is very many 

 times greater than that expressed by the momentum which the 

 pound of water would have on arriving there without resistance; 

 and with regard to the six footqwunds, we have seen that the 

 work is no more comparable to them than a surface is to a 

 solid. 



Again, if we suppose the gradient upon which gravity acts to 

 be only one foot in the distance instead of six, and the resist- 

 ance lo the pound of water to be as the velocity, then the water 

 would move with only one-sixth of the velocity in the other case, 

 and the water would be six times as long in reaching the parallel 

 of 60", but the energy of the action of gravity would be only one- 

 sixth as much, and henc; the work would be the same, being 

 carried on six times as long in the latter case with one-sixth of 

 the energy. But then the same work would be represented by 

 one foot-pound, if that is a true unit of the work instead 

 of six. In this case also the deflecting force eastward would 

 be only one-sixth as much, but the time being six times as long, 

 the same amount of work would be done, and this would be 

 sensibly the same as that which would be required to give the 

 pound of water a velocity of about 760 miles, as Mr. Croll has 

 it, but really double that amount. 



I am well aware that in the action of machines in which 

 force is balanced against force, and consequently the times of 

 action are the same, the amount of work may be expressed 

 by the forces multiplied into the spaces through which they 

 act ; but in all cases in which the times differ, the amount 

 of work cannot be expres^ed by any unit into which the element 

 of time does not enter. The amount of work required to pro- 

 duce a velocity of 760 miles per hour is a function of the time, 

 and proportional to the time where the force is constant, and 

 cannot be measured by foot-pounds. 



With regard to the argument ba=ed upon M. Dubuat's experi- 

 ment, the matter briefly and fairly stated stands thus : according 

 to the experiment, water will not flow unless acted upon by a 

 force equivalent to that of gravity upon a given gradient, whicli 

 makes the force required to move it about fifteen times greater 

 than the horizontal componer.t of the moon's disturbing force 

 which produces the tides. But this force of the moon does move 

 the water of the ocean, and therefore M. Dubuat's experiment is 

 not applicable to water of great depth as of the ocean, and 

 the argument fails. It is tiue, as Mr. Croll states, that the 

 two cases of motion are somewhat different; in the case of the 

 tides the w^ater from top to bottom flows in the same direction, 

 while in the other the upper and lower strata flow in contrary 

 directions, and the resistances to the lower motions are no 

 doubt greater. Six of the nine feet therefore of i\Ir. Croll's 

 gradient should probably be given to the lower currents, and 

 only three to the upper ones. But Mr. Croll admits that in 

 the case of the tides a gradient of one inch is sufficient to move 

 thewater. A gradient of three feet only, therefore, ought to be 

 sufficient to move the upper half, which would correspond some- 

 what to the case of tides in an ocean of half the depth. The 

 observations of Col. Graham show that the water of Lake Michi- 

 gan, .about 700 feet deep, readily yields to the moon's disturbing 

 force which causes a tide at Chicago with a range of nearly two 

 inches. 



Cambridge, Mass., Sept. 7 Wm. Ferrel 



Spectral Nomenclature 

 It seems almost absurd that a subject of such interest, and, as 

 I think, importance, as that of Spectral Nomenclature should 

 be discussed from opposite sides of the globe alone ; so it may 

 be hoped that it will not have been allowed to end with Prof. 

 Young's remarks upon the one or two points in which he differs 

 from me, but will have been taken up by others at a less distance. 

 There is a great deal more to be said about it ; but probably I 

 should not have troubled you again just yet but for the obligation 

 I feel to disclaim credit which he gives me for what is not mine. 



