286 Prof. Challis on the Principles of Hydrodynamics. 



there results 



du dv dw_dY n l\ 



d^'^ dy ^ dz ~ ds "*■ 'Vr ^ r')' 



Hence equation (7.) readily follows by means of equation (2.). 



This demonstration has not required the use of equation (3.). 

 It must, however, be observed, that we should not have been 

 justified in reasoning with the unknown function \, unless that 

 equation had shown it to be a discoverable quantity; nor could 

 we have concluded from the above demonstration (what is evi- 

 dent from the other demonstration), that the equation (7.) is 

 perfectly general, if the general equation (3.) had not been pre- 

 viously demonstrated. 



The propositions hitherto proved apply both to compressible 

 and incompressible fluids. Let us now suppose the fluid to be 

 incompressible. Then since p is constant, the equation (2.) 

 becomes 



du dv dw ^ /Q V 



di + d^+dF=^ ^^-^ 



Hence equation (7.) becomes 



^-<l-lr)=0 ,9.) 



We are now prepared to solve the following problem. 



Proposition IX. It is required to determine the law of action 

 of the parts of an incompressible fluid on each other. 



By the proof of Prop. VII. it appeared, after satisfying the 

 condition of integrability of udx + vdy + ivdz independently of 

 any particular case of motion, that the orthogonal trajectories of 

 the sm-faces of displacement were at each instant straight lines. 

 Let us therefore, in accordance with this result, substitute dr for 

 ds in the equation (9.), and integrate along the trajectory. We 

 shall thus obtain 



V= ^ (10.) 



rr 



Hence the velocity at any point of the trajectoi-y is given, if the 

 velocity and radii of curvature of the surface of displacement at 

 a given point of it be given. We have thus arrived at a definite law 

 of variation of the velocity prior to the consideration of any case 

 of disturbance, and may therefore conclude that equation (10.) 

 expresses the law of action of the parts of an incompressible fluid 

 on each other. 



If it be objected to this reasoning that it depends on the gra- 

 tuitous assumption that X is a fimction of yjr and t, I answer that 

 the assumption, having no reference to any arbitrary case of dis- 

 turbance, is precisely of the kind that the nature of the iuves- 



