I: INSTANCE. 



.. of one kind. [MATERIALS, STUKSOTH or.] Again, when 

 body U made to more on another, the inequalities of the surface* uf 

 both eratto a rasbtutte of different kind. [Kmcrio*.] When 

 mom* in a fluid, the inertia of the fluid particle* displaced by it 



a third kind of raaistanoe. 



Thin lait brmnch of the subject of resistance* hai already been in 

 part couatdsred under HTDBODT.XAMU-*. In that article there U giren 

 a general expression for the moaiure of the resistance made by a fluid 

 against a plane turface which U either perpendicular or inclined to 

 the direction of the motion, together with a few results of expert 

 meat* on the remittance* experienced by bodies of various forms 

 and length* in moring through water. "To* relation! between ipacea 

 an<l time* in the vertical aacent and descent of bodies when acted on 

 by grarity and reacted by a fluid, are given in the article PBOJEC 

 TILES, TRKOBT or ; and, for the pressure against a cannon-ball moving 

 in air, see the article Ouxxnr. 



In investigating the resistances of fluid* again*t bodies moving in 

 them, it U customary in elementary writing*, for the sake of aim pi i 

 city, to consider the particles of fluid as unconnected with each other 

 by contact or by any law of attraction, ao that, when struck, their re- 

 actions may be considered a* taking place perpendicularly to the strik- 

 ing surface of the moving body, whatever be the position of this 

 surface with respect to the direction of the body's motion, and after 

 the impact their action i* supposed to cease. Such are called discon- 

 tinuous fluids, and in these the motion produced in the particles by 

 the collision is the measure of the resistance. Newton shown 

 C Principia,' lib. ii.. prop. 85) what would be the resistance exp< 

 by a cylinder moring in the direction of its axis in a discontinuous 

 fluid ; the cylinder and particles uf fluid being elastic, BO that the 

 latter on being struck are reflected back with a velocity double the 

 Telocity of the cylinder ; and he explains that, if the particles of fluid 

 are not reflected, but are moved forward by the cylinder witli a 

 Telocity equal to its own, the resistance is but half the former. But 

 this hypothesis is far from being conformable to the constitution 

 of fluid bodies in nature, the particles of these being connected 

 by mutual actions. The elastic fluids, as air, at any place in tin- 

 atmosphere are always in a state of compression from the weight of 

 the column vertically above that place ; and the particles of non- 

 elastic fluids, as water, exert in every direction pressures which d. ].,,. I 

 upon the distances of the particles below the surfaces of the fluid in 

 the vend, river, or ocean. In passing through a fluid of this kind 

 (called a continuous fluid) a body strikes only the fluid particles which 

 are nearest to it ; these strike those beyond, and so on ; and Newton 

 proves (lib. ii., prop. 85, schoL) that in this case'the resistance to a 

 cylinder is only half the last-mentioned resistance, or one-fourth of tin- 

 first. 



In all these resistances however it is supposed that the particles on 

 being struck are repelled |>eri>endicularly to the front of the moving 

 body ; but, in fact, the particles of the fluid are in part repelled from 

 the front in oblique directions, and, on account of the compressed 

 state of the surrounding fluid, these particles not being able immedi- 

 ately to escape laterally, there is produced in front more or less con- 

 densation, and consequently an increase of resistance. The pressure 

 of the fluid against the sides of the moving body create* also a resist- 

 ance from friction ; and when the velocity is very great, the fluid not 

 falling towards the hinder part nf the body so fast as the latter moves, 

 the pressure there which would serve to counterbalance the resistance 

 in front, is in part or wholly removed. On these accounts it is that 

 military projectiles are subject to such vast retarding forces. It is 

 computed that a 24-pounder ball experiences a resistance equal to 

 800 Iba. when its velocity is equal to 2000 feet per second. Like 

 effects take place in the movements of boats and ships; when the 

 Telocity is great, the water accumulates in front, and flowing off from 

 thence obliquely, it carries away some from the sides, and, causing the 

 surface of that which is near the stern to be rather lower than the 

 general level, it there produce* a diminution of pressure, while there is 

 an excess in front on account of the accumulation. 



In rder to find the pressure of a fluid against a body which is ter- 

 minated in front by a curve surface, an expression must be obtained 

 (by means of the equation of the surface) for the area of an el.-m. -n- 

 tary portion of that surface, and this must be nriltij.li.-d by the cube 

 of the sine of its inclination to the line of motion. The product 



being multiplied by !!l [HTDBODTNAMICS], and the whole inte- 



tg 



grated between the proper limits, the result will express the required 

 resistance. 



Again, in investigating the motion of a body on an inclined plane 

 when resisted by friction and the pressure of the atmosphere, the 



general equation of motion =ya may be employed. Here 

 dl* ftp 



is the space described in the time I, is the Telocity acquired in th.- 



at 



matt time, and - is the differential expression for accelerativc or 

 at* 



retardative force. If the body were to descend vertically, y, the force 

 of gravity ( 3lM7 feet), would alone be the force producing the 

 notion ; and the equation, being integrated, would give the i 



between the spaces described and the times of description when the 

 body descetxl* or ascends in a resisting medium. In the first of these 

 ease*, y should be positive, and in the second negative. In order to 

 .vl.ipt the equation to the descent of a body on an inclined 

 let be the inclination of the plane to the horizon ; then ;/ xin. 6 would 

 reprewnt the aocelerative force on the plane if there were no friction. 

 But since friction is proportional to the pressure (=g cos. ) on th.- 

 plane, and U independent of the Telocity, let A be put for the coeffi- 

 cient of friction and represent a fractional part of the pressure ; then 

 we shall have hy cos. t for the retardation produced )>y i 

 a is the coefficient of the resistance due to the pressure of th.- 

 atmosphere; it depends on the form and magnitude of the moving 

 body, and not on its weight ; and the resistance is supposed to be 

 proportional to the square of the velocity. Thus the above equation 

 M M 



? =g sin 9-hy ootS-a; 

 d(* d{* 



or, since the two first terms of the second member are constant, repre- 

 senting them by A, it become* 1 = A a . Integrating this equa- 



d(* at* 



ti -n l.y successive approximations, or otherwise, we obtain in terms of 



/ the values of (the velocity) and of $ (the distance on the plane), 



SB 



either when the body sets out from a state of rest, or when it 

 set* out with any given initial velocity. From these values, by 

 means of the data obtained from good experiments, the values of A 

 and a might be found; and thus the effects of friction might be 

 obtained separately from those which are due to the resistance of 

 the air. 



The formula; which are now generally received as expressing the 

 resistance* to which waggons moving upon railway* are exposed are as 

 follows: They are of two classes; the one normal, and the other 

 accidental ; the one susceptible of a priori calculation, the other 

 depending upon the state of the road, the action of the wind, Ac., or 

 upon conditions of an essentially variable nature. Of the noria;il 

 causes of resistance, there are three kinds : 1, the friction of the axles 

 in the boxes : 2, the friction of the whet-In upon the rails ; and 3, the 

 resistance of the air, supposed to be in a state of repose, to the advance 

 of the train. The level of the surface of the roadway naturally affects 

 the numerical calculations of the various conditions thus specified; 

 but, as was before said (under RAILWAY), the introduction oi 

 expansion gear into locomotive machinery has so modified the powers 

 of that class of engine, that the importance of the precise value of the 

 resistances to be overcome has of late been materially diminished. 

 With an engine whose powrrx can be increased at will, and I 

 instantaneously, variable resistances are really matter* of very little 

 moment. The formula; for this class of resistance are extracted from 

 Pcrdonnet's ' Traitl Elementaire des Cheming de Fer.' 



Taking into account, firstly, the cose of a waggon moving in plain 

 and on a dead level, it is evident that the horizontal movement must 

 produce a friction of the bearing surface of the axle-boxes upon the 

 axles themselves, which will be proportional to the pressure (or to the 

 weight of the carriage, minus that of the wheels and axles), and will 

 vary with the state of the bearing surfaces, but independently of their 

 own state. If, then, the pressure upon the axle-boxes be represented by r, 

 and the coefficient of friction by / (it will, in fact, be regulated by "the" 

 nature of the bearing surfaces, their smoothness, aud the nature of the 

 lubricating material), the friction of the boxes on the axles will be 

 represented by /P. Then, calling B the radius of the wheels, and r 

 the radius of the bearing of the axle, every revolution of the wheels 

 will cause the waggon to advance a distance of 2 r n, and every point 

 of the bearing a distance of 2r ; so that whilst the waggon advances 

 through a distance = 1, the bearings of the axles will have slid over a 

 surface _ZT =H. The value of the friction of the bearings will then 



*. V H II 



be, for the same distance traversed, /r -. 



Th.- friction of the wheels against the rails is a friction of the kind 

 known as a rolling friction, and as such it is generally considered to be 

 proportional to the pressure, and variable according to the nature of 

 the surface* in contact, but independent of the area of those surfaces, 

 and of the speed of the motion. Or, calling /) the weight of the wheel* 

 and axles, the total pressure or weight will be v+f, and /' the 

 coefficient of friction, the expression of this description of friction will 

 then become f(r 4- />). Strictly speaking, the value of/' would ,i 

 on the sue of the wheels ; but as, practically, the wheels of railway 

 carriages are of the same diameter,/' may Ixt considered to be a constant 

 quantity. 



\Vh.-n a body moves in a discontinuous fluid in repose, the resistance 

 it meet* with U proportional to the square of the velocity; to the area 

 of the lection of the bodv moved, taken normally to the direction of 

 the movement: it is less in proportion to the length of that body in 

 the direction of the movement ; and if two surfaces, covered one by 

 the other, move in the same direction, the resistance of the covered 

 face will be equal to a fraction of the face immediately exposed to the 

 air; and th<> cmaller the interval between the two faces, the less will 



