308 Mr. stokes, ON THE FRICTION OF FLUIDS IN MOTION, 



but no value of C different from zero will allow Q to vanish when < = 0, whereas by hypothesis 

 it does vanish; hence Q = 0; but Q is the sum of 71 quantities which evidently cannot be 

 negative, and therefore each of these must be zero. Since then a)|...(u„ would have to be equal 

 to zero for all values of t from to t even if they satisfied equations (26), they must d fortiori 

 be equal to zero in the actual case, since they satisfy equations (25). Hence there is no value of 

 / from to T at which any one of the quantities tu, ...ai„ can begin to differ from zero, and 

 therefore these quantities must remain equal to zero for all values of t from to T. 



This lemma might be extended to the case in which « = os , with certain restrictions as to 

 the convergency of the series. We may also, instead of the integers 1, 2...n, have a continuous 

 variable a which varies from to a, so that to is a function of the independent variables a and t, 

 satisfying the differential equation 



d 



at Jo 



where \|/(a, 0, t) does not become infinite for any value of a from to a combined with any 

 value of t from to T. It may be shown, just as before, that if o) = when t = for all values 

 of a from to a, then must oi = for all values of t from to T. The proposition might be 

 further extended to the case in which a = 05 , with a certain restriction as to the convergency of the 

 integral, but equations (25) are already more general than I shall have occasion to employ. 



It appears to me to be sometimes assumed as a principle that two variables, functions of 

 another, t, are proved to be equal for all values of t when it is shown that they are equal for a 

 certain value of t, and that whenever they are equal for the same value of t their increments for 

 the same increment of t are ultimately equal. But according to this principle, if two curves 

 could be shown always to touch when they meet they must always coincide, a conclusion 

 manifestly false. I confess I cannot see that Newton in his Principia, Lib. i. Prop. 40 has 

 proved more than that if the velocities of the two bodies are equal at equal distances, the 

 increments of those velocities for equal increments of the distances are ultimately equal : at least 

 something additional seems required to put the proof quite out of the reach of objection. Again 

 it is usual to speak of the condition, that the motion of a particle of fluid in contact with the 

 surface of a solid at rest is tangential to the surface, as the same thing as the condition that the 

 particle shall always remain in contact with the surface. That it is the same thing might be 

 shown by means of the lemma in this article, supposing the motion continuous ; but independently 

 of proof I do not see why a particle should not move in a curve not coinciding with the surface, 

 but touching it where it meets it. The same remark will apply to the condition that a particle 

 which at one instant lies in a free surface, or is in contact with a solid, shall ultimately lie in the 

 free surface, or be in contact with the solid, at the consecutive instant. I refer here to the more 

 general case in which the solid is at rest or in motion. For similar reasons Poisson's proof of the 

 Hydrodynamical theorem which forms the principal subject of this section has always appeared 

 to me unsatisfactory, in fact far less satisfactory tlian Lagrange's. I may add that Poisson's 

 proof, as well as Lagrange's, would apply to the case in which friction is taken into account, in 

 which case the theorem is not true. 



13. Supposing ju to be a function of p, , the ordinary equations of Hydrodynamics 



J \P / 



*'^ dx ^ Bt' dy -"^ Bt' dz '^~Bt ^ ^' 



The forces X, Y, Z will here be supposed to be such that Xdx + Ydy + Zdx is an exact 

 differential, this being the case for any forces emanating from centres, and varying as any functions 



