106 The Rev. J. Challis on the Rectilinear Motion of Fluids, 



Now, if the coordinates be supposed to vary from one point 

 to another of a surface of displacement, from what is proved 



above, dV= 0. Alsod.-r-= ~dt ~7rA udx + V(l y + wdz) 



= 0, because for a surface of displacement udx + vdy + wdz 

 = 0. Hence, dividing by ds the increment of space, 



qds \ ds ds ds/ 



The left-hand side of this equation is the effective accelerative 

 force in any direction perpendicular to that of the motion. 

 As this force vanishes, the motion must be rectilinear. 



It follows from this reasoning that the sole and the neces- 

 sary condition of the rectilinear motion of a fluid is, that udx 

 + vdy + it)dz be an exact differential of a function of three 

 independent variables. 



It has been argued by Lagrange in the Mecanique Analy- 

 tique, that udx + vdy + ivdz is an exact differential when 

 the motion begins from rest, and again, when the motion is 

 so small that the squares and higher powers of u, v, and id 

 may be neglected. These propositions are inserted in the 

 edition of Poisson's Traite de Mecanique of 1811, but are 

 omitted in that of 1833. In the Memoirs of the Academy of 

 Paris (tome x. 1831), Poisson considers a problem in which 

 that condition is not fulfilled, though the motion is small. 

 Against the former of the above propositions it may be urged 

 that when u = 0, v = 0, id = 0, it cannot be asserted of u d x 

 •4- vdy + tad si either that it is integrable or that it is not 

 integrable ; and against the latter, that the integrability of the 

 quantity in question is in no respect dependent upon the mag- 



nitudes of u, v and id- For example, V . d x + V . - — dy 



Z —- C £ — * C 



+ V . dz, is as far from being integrable when V is a 



very small quantity, as when V is large. On this account, 

 the cases of fluid motion in which udx + vdy + ivdz is an 

 exact differential must be determined by considerations inde- 

 pendent of the magnitude of the motion, as I have done in 

 this communication. 



To prevent misapprehension on this subject I may also 

 remark, that it is possible to assume at pleasure values of w, 

 v and iv, which will satisfy the equation of continuity and 

 make udx + vdy + id d z integrable, and at the same time 

 give a curvilinear motion. For example, if u = m x, v = — my 

 and id = 0, and the fluid be incompressible, each particle moves 



