30 Professor Kelland's Note on Fluid Motion. 



under which the equations of fluid motion can be solved. 

 Whilst interest is awakened on the subject, it may not be 

 deemed utterly unimportant to offer a few remarks on the 

 general question, especially as any peculiarity in the mode of 

 proceeding, however valueless in itself, may serve as a hint 

 to guide or incite others to the most important investigations. 

 The question before us appears to me to be this — What 

 new conditions must we introduce, or what transformations 

 must we effect, in order that the four equations of fluid mo- 

 tion may be reduced to four other equations, each containing 

 the differential coefficients of only one quantity ? Before this 

 question can be answered, at least before we can set about 

 introducing any new conditions, it appears requisite to an- 

 swer another question — Are there any necessary conditions ? 

 Of course the answer is in the affirmative. The equation of 

 continuity is one. But it is not the only one; for unless the 

 pressure and velocities are discontinuous quantities, the equa- 

 tions deduced by the application of D'Alembert's principle 

 must be statical equations, depending on the time only in as 

 far as the velocities depend on the time. Hence the relations 

 which would exist amongst the differential coefficients of p, 

 were the fluid at rest, must exist when it is in motion ; that is, 



d? p d 2 p _ 



d x dy ~ dy d a? 



These, then, are equations of condition ; the bearing of 

 which ought to be examined previous to the introduction of 

 any new conditions. They will serve, in some cases, to show 

 what new hypotheses are admissible, and, in all, to detect 

 those which are not. 



It is not my intention to enter fully into this subject in my 

 present communication. I shall content myself with offering 

 a few remarks on the results of the mode of proceeding which 

 I have indicated, as applied to the motion of incompressible 

 fluids acted on by gravity only. 



By inclosing within brackets the complete differentials with 

 respect to x, y, z and t, we obtain the following sets of equa- 

 tions : — 



(1.) 



