40 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



/da\ 

 Hence - a {^) . .N W + f) = 0, and N = ^^^. 

 But by the equation a? - if - a' = 0, (since x, y, and a vary simultaneously with the position 

 of the element,) we have ^ ( J) " ^ (J) " « (S) = °' ""'^ (S) = '' = '"^' (S) = " = " '^^^^ 

 Hence a (• — ) = m{ii^ + i^), and consequently A?" = — . This value makes —dx + — dy an 



exact differential, and the equation (3) is therefore verified. We have now to see in what 



dV ds 

 manner equation (8) is verified. This equation for the instance before us becomes ^~ + — = 0, 



V r 



r being the radius of curvature of the curve of displacement at the point a)y, and ds the increment 



{ay + «")t i3^-\-'if\\dx 

 of the line of motion at the same point. Hence r = — — ; — ^-7- , and ds = . There- 

 in'' — y' x 



fore . — = 0. This equation cannot be integrated unless y is eliminated by means 



V a^ + y"- X 



of the equation xy = c' ; that is, it can be integrated only along a line of motion. The dynamical 

 equation must therefore be integrated in the same manner, and the arbitrary quantity to be 

 added is a function of co-ordinates as well as the time. The fluid must be conceived to be 

 included between two hyperbolic surfaces indefinitely near each other. This explains the contra- 

 diction met with in Art. 3. 



y 



Ex. 2. Let the equation of the surfaces of displacement be — tan"' - = 0. Putting therefore 



a 



/for - tan"'— , we have 



de'Kdtj \dtj' dx ai' + y'' ^" dy or' + y'' 



Now since y = x tan 9, the motion is evidently parallel to the plane of xy in concentric circles about 

 a fixed axis. Hence at any distance r from the axis F = r I — I . Consequently N = (.r^ + y^) 



f(.t) 

 ~ — Vr. Therefore if F = , we have N a function of t, and vdx + vdy an exact differential, 



T 



although the motion is curvilinear. This is the case of motion alluded to at the end of Art. 7. 



Ex. 3. Let it be required to determine whether in an incompressible fluid the surfaces of 

 displacement can be concentric spherical surfaces, the centre of which is always on the axis of 

 X, and at the same time the motion be such that a given particle in successive instants is at 

 the same distance from the common centre. 



Here if a = the co-ordinate of the centre, and a = the radius of any surface of displacement, 

 we have /= (a? - aY + y" + «- - a^ 



^^ df /da\ /da\ 



df (da\ (da\ 



Ta' Idjj =''' ''''=^"'" ^^ hypothesis (-J =0. 



