430 The Rev. Prof. Challis on the Motion of Fluids. 



for February 184'4', p. 94). They are not even mentioned 

 in Poisson's Treatise on Hydrodynamics (2nd edition), 

 which may be regarded as the standard woi'k on this subject; 

 yet they are made use of to the present day without scruple, 

 apparently on the mere authority of Lagrange's assertion. 

 Thus it has happened that an essential step has been taken in 

 the application of the general equations without any reasons 

 to justify it. The new equation which I have discovered is 

 absolutely necessary for treating questions in hydrodynamics, 

 for which udx + vdy + nxxlz is not integrable without u factor; 

 and, what is very important, it furnishes a criterion for distin- 

 guishing the cases in which that quantity is integrable of itself. 

 For, since by hypothesis, 



U 1 'V r 1^ » / 7 IS 



— dx-\ aj/ -\ az = (a\f/), 



AAA 



it follows that 



rfj/ d^ rfvl/ 



And by substituting in (3.), we have 



which is the additional equation necessary in consequence of 

 the additional variable A. Now if s be a line drawn at a given 

 instant in the direction of the motion of the particles through 

 which it passes, and V be the velocity at a point xy z of this 

 line, we shall have 



ds^ \dx^ dy^ dz^/ 



and the foregoing equation becomes 



dt ds^ 



Consequently, if A be a function of the time only, which is the 

 case when 7idx + vdy + wdz is integrable without a factor, this 

 equation gives by integration vf/, a function of s and / ; at the 



same time V or A -7 is a function of s and /. The criterion 

 ds 



we thus arrive at is, that ^ and V are both functions of the 

 same variables s and t. I have proved in the Cambridge 

 Philosophical Transactions (vol. viii. part i. p. 36\ that this 

 condition is not fulfilled unless the motion be rectilinear. 



The reason of the contradictory result in the instance ad- 

 duced above will now be apparent. It is there first supposed 

 that udx + vdy is an exact differential, and then forms are 

 given to the functions F and^^ which, as they apply to curvi- 



