CALCULUS OF PARTIAL DIFFERENCES. 41 



which prevents them not from obeying an external impulse, it 

 is manifest that the principle may be applied. Thus, if a 

 fluid contained in a vessel of any shape be conceived divided 

 into layers perpendicular to the direction of its motion, and if 

 v represent generally the velocity of the layers of fluid at any 

 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 + dv 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. 

 chap. 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 remained after all his efforts incapable 

 of a complete solution, partly owing to the inherent difficul- 

 ties of the subject 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 theoiy of 

 the Eesistance 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 ; namely, that it is a 

 body composed of very minute particles, separate from each 

 other, and capable of free motions in all directions. He 



