TRANSACTIONS OF THE SECTIONS. 213 



tides wliicli it .affects, and as these in turn, when put in motion, affect other parti- 

 cles outside tliem, the remainder of the force is transmitted to those particles. 

 Hence it becomes desirable to conceive an arrangement under which the force 

 would be transmitted entire, instead of being partly absorbed in momentum, and 

 to trace the law of transmitted motion which would correspond with this. 



(4) Such a state must be conceived to exist if we imagine two parallel planes, of 

 infinite extension each way, having the intervening space filled with Huid, and 

 having possessed for an infinite period equal edgeways velocities (say v) in oppo- 

 site directions. The trnnsmission of motion and of force in the intervening fiuid 

 must have become established and permaneiit, and the following propositions woidd 

 seem true respecting it. 



(a) Each plane must be experiencing throughout its surface a definite frictional 

 force per square foot, acting in the direction of the plane, equal and opposite in the 

 two planes, and the force must be transmitted statically from plane to plane by the 

 state of motion in the intervening fiuid. 



(b) Along the imaginaiy plane which bisects the intervening space, the particles 

 of fluid must be stationary, since they are similarly situated with respect to the two 

 equal opposite forces and motions. 



(c) Between the imaginary central plane, and each of the two moving planes, 

 the particles must possess a graduated motion corresponding with that of the 

 nearer plane. 



If we further imagine the whole intervening space to be subdivided into laminfe 

 or layers of equal infinitesimal thickness, say 1, 2, 3, . . . n, it would seem that, to 

 transmit the definite force vmchanged from layer to layer. No. 1 must be gliding 

 past Xo. 2 exactly as fast as No. 2 past No. -3, and so on throughout, from one 

 outer plane to the other, for these are to be judged as adherent to the fluid, as the 

 fluid to itself (see prop. 1) ; .since that supposition, exclusively, secures throughout 

 an identity in the relative motions of similarly placed contiguous particles : if we 

 were to suppose different gradations of velocity in relation to lateral space, or 

 thickness of layer, in different parts of the series of layers, it would follow that dif- 

 ferent relative motions of similarh' placed contiguous particles would develope the 

 same amount of force. 



The character of this graduated transition of velocity may be best explained by 

 observing that if at any moment a straight filament were laid at right angles 

 across the interval between the planes, and were to accept the motion of the 

 particles which it traversed, it would remain straight, while it continued to 

 describe a growing rectilinear angle, the angle being as the time, and the centre of 

 motion being the centre of the space between the planes where the particles are 

 .stationary. 



The conception of this angle will be relied on as important in the course of the 

 investigation, and it will, for convenience, be termed the " filamental" angle (cf)), 

 and will be taken account of as described in some definite infinitesimal interval 

 of time, At. 



Bearing in mind the graduated transition of velocity, it follows that if we were 

 to substitute for any one of the interior layers a plane similar to the original planes, 

 assigning to it the velocity of the layer for which it is substituted, we might re- 

 move the fiuid external to it on one side without in any degree altering the state 

 of relative motion (as expressed by the filamental angle) and of force, in the 

 layers of fiuid which remained. 



The absolute velocity of this plane might be less in any assignable ratio than that 

 of the original planes, j'et it would expeiience the same resistance per square foot 

 as they ; and it follows that the frictional force experienced by a surface moving 

 past the particles of a fiuid, is independent of the absolute velocity of the plane, so 

 long as the filamental angle is imchanged. If, retaining the original planes and 

 the space between them unaltered, we assign to them an established velocity, say 

 7)1 times as great as before, the filamental angle will also have become m times 

 as great, expressing the circumstance that the relative motion of similarly placed 

 contiguous particles is increased in that ratio, and that a condition appropriate to 

 a difference of frictional force has been established. The force of fluid friction 

 must be regarded as governed, not by the absolute velocity of the moving surface, 

 but by its velocity as related to that of contiguous particles, expressed in terms 



