296 Prof. Chaliis oft a new Equation in Hydrodynamics, 



for obtaining (1.). It must, however, be observed, that Pro- 

 fessor Tardy assumes the existence of the function A, without 

 which it would not be possible to arrive at the equation (1.), 

 which is admitted to be a true equation, and yet he has not 

 shown in what way that function may be found. There 

 can be no doubt that it is a discoverable quantity ; and since 

 the number of recognized equations is only equal to the num- 

 ber of the other unknown quantities, it follows that another 

 equation is necessary to make up the requisite number of 

 equations. Consequently the necessity for a new equation 

 results from Professor Tardy's reasoning. 



Respecting equation (1.) I have further to remark, that it 

 is not, as Professor Tardy states it to be, merely an analytical 

 transformation of the equation 



dp d.pu d.pv d.pw _ 



Tt '^ ~d^ ^ 'df'^ IT -^' (^-^ 



for in that case such a function as \ would not be required 

 for deducing it. It differs from (3.) by containing an explicit 

 expression of the principle, that the directions of motion in 

 any given element at any time are normals to a continuous 

 surface, which principle is tacitly adopted when we assume 

 that the differential calculus is applicable to the consideration 

 of fluid motion. The equation (1.) cannot be obtained without 

 expressly taking account of surfaces of displacement, the con- 

 sideration of which has usually been omitted by writers on 

 hydrodynamics. 



I admit also that the reasoning which Professor Tardy is 

 unable to appreciate, viz. that by which I derive the equation 

 (2.) from fundamental principles, requires further elucidation; 

 and I will now endeavour to place it in a clearer light. Let 

 it be granted that the directions of motion in any given ele- 

 ment of the fluid at any time are normals to a continuous sur- 

 face. Then, 4' being an unknown function of the co-ordinates 

 and the time, it follows from this principle that «, v and w 



are respectively in the proportion of ^, ■^, and -^; or, A 



being another unknown function of the co-ordinates and the 

 time, that 



d'h d-h dyb 



"=^S' ''=^^' '*'=^s- 



Hence 



{d^)= -dx+ -dj/+ -dz. 



AAA 



Thus the right-hand side of this equality is integrable in con- 



