410 D'ALEMBERT. 



instant, and d v the small increment of that velocity, 

 which may be either positive or negative, and will be 

 different for the different layers, v vo will express the 

 velocity of each layer as it takes the place of that 

 immediately below it ; then if a velocity + dv alone were 

 communicated to each layer, the fluid would remain at 

 rest. ('Traite de Fluides/ Liv. ii. ch. 1. Theor. 2.) 

 Thus the velocity of each part of the layer being taken 

 in the vertical direction is the same, and this velocity 

 being that of the whole layer itself, must be inversely as 

 its horizontal section, in order that its motion may not 

 interfere with that of the other layers, and may not 

 disturb the equilibrium. This, then, is precisely the 

 general dynamical principle already explained applied to 

 the motion of fluids, and it is impossible to deny that 

 the author is thus enabled to demonstrate directly many 

 propositions which had never before been satisfactorily 

 investigated. It is equally undeniable that much re- 

 mained after all his efforts incapable of a complete solu- 

 tion, partly owing to the inherent difficulties of the sub- 

 ject from our ignorance of the internal structure and 

 motions of fluids, and partly owing to the imperfect state 

 in which all our progress in analytical science still has 

 left us, the differential equations to which our inquiries 

 lead having, in very many cases, been found to resist all 

 the resources of the integral calculus. 



This remark applies with still greater force to his next 

 work. In 1752, he published his Essay on a new 

 theory of the Resistance of Fluids. The great merit of this 

 admirable work is that it makes no assumption, save one 

 to which none can object, because it is involved in every 

 view which can well be taken of the nature of a fluid; 



