Prof. Challisow a new 'Equation in Hydrodynamics. 287 



or, neglecting quantities of the second order, 



- mlt {M' q' - \ q) - m\ q (^ + ~\'^rlt. 



And this increment is also equal to ?« 8 ?• 8 q. Hence by equa- 

 ting the two values and passing from differences to differen- 

 tials, we have the equation sought, viz. 



^s + '^ + v,(l+i)=o. • 



dt dr ^ \r t' / 



But this proof does not possess more generality than that 

 which, as mentioned above, is derivable from the equations of 

 xhe Mecanique Analytiqiie; because it is here assumed that 

 the Jbcal lines to which the lines of motion are directed, are 

 fixed in space; in other words, that the motion is rectilinear. 

 To prove that the same equation is arrived at when the focal 

 lines change their positions so that the motion of the fluid par- 

 ticles is curvilinear, requires considerations which I have en- 

 tered into in the Numbers of this Journal for December 1 840 

 and June 184'1. In the method of obtaining equation (6.), 

 employed in the present communication, the distinction be- 

 tween the two cases can only be made by the factor N. Ac- 

 cording as N is a function of t only, or a function of t and the 

 coordinates, the motion is rectilinear or curvilinear ; and hence 

 we may draw the inference, that motion in ajluid is rectilinear 

 or curvilinear accmding as u dx + vdy-\-w d ;: is integrahle of 

 itself or by a factor. 



The truth of this inference appears also from the following 

 argument. When iidx 4- vdy + w d z is assumed to be an 

 exact difiFerential, the resulting partial differential equation in- 

 volving the variables <^, x, y, z and t, is of the second order, 

 and its integral contains two arbitrary functions. But when 

 that quantity is made integral by a factor, the resulting equa- 

 tion, as 1 have shown in the communication to the February 

 Number, rises to the third order, and its complete integral 

 consequently contains three arbitrary functions. The third 

 function has reference to the variation of density which must 

 exist at each instant at a surface of displacement when the 

 motion is curvilinear, and which disappears when the motion 

 becomes rectilinear, because in this case the surface of dis- 

 placement coincides with a surface of equal density. 



The discussion I have now gone through may suffice to 

 establish the truth and importance of equation (3.), which ap- 

 pears to be absolutely necessary for giving to the differential 



