1871.] 



Uniform Flow of a Liquid. 



287 



sistance in what would otherwise have been the motion of each individual 

 molecule of the liquid so disturbed. 



This problem, however, is by no means so difficult as the other. There 

 is, indeed, a case in which it admits of solution. It is that of a liquid 

 flowing from a reservoir, in which its surface is kept always at the same 

 level, through a circular pipe which is perfectly straight, and of the same 

 diameter throughout, and of a uniform smoothness or roughness of internal 

 surface, and always full of the liquid. The liquid would obviously in such 

 a pipe arrange itself in infinitely thin cylindrical films coaxial with the pipe, 

 all the molecules in the same film moving with the same velocity, but the 

 molecules of different films with velocities varying from the axis of the pipe 

 to its internal surface. The direction of the motions of the molecules of 

 such a liquid being known, and all in the same film moving with the same 

 velocity, which velocity is a function of the radius of the film, and the law 

 of the resistance of each film to the slipping over it of the contiguous film 

 being assumed to be known, as also the head of water, it is possible to 

 express mathematically 



(1st) the work done per unit of time by the force which gives motion to 

 the liquid, and 



(2nd) the work per unit of time of the several resistances to which the 

 liquid in moving through the pipe is subjected, and 



(3rd) the work accumulated per unit of time in the liquid which escapes— 

 and thus to constitute an equation in which the dependent variables are 

 the radius of any given film, and the velocity of that film. This equation 

 being differentiated and the variables separated, and the resulting differen- 

 tial equation being integrated, there is obtained the formula 



250 r 



V: 1 ' 



where v is the velocity of the film whose radius is r, and v that of the 

 central filament, and I the length of the pipe — the unit of length being one 

 metre, and of time one second. 



The method by which the author has arrived at this formula is substan- 

 tially the same as that which he before used in a paper read before the 

 Society on the " Mechanical Impossibility of the Descent of Glaciers by 

 their weight only," and which he believes to be a method new to me- 

 chanical science. It was indeed to verify it in its application to liquids 

 that he undertook the investigations which he now submits to the Society, 

 which, however, he has pursued beyond their original object. 



The recent experiments of MM. Darcy and Bazin* have supplied him 

 with the means of this verification. These experiments, made with ad- 

 mirable skill and precision, on pipes upwards of 100 metres in length, and 

 varying in diameter from m, 0122 to m, 5, under heads of water varying 



* Kecherches Exp6rimentales relatives au mouvemortfc de l'Eau dans les Tuyaux, 

 par H. Darcy: Paris, 1857. Recherches Hydrauliques, par MM. Darcy et Bazin : 

 Paris, 18G5. 



