300 Prof. Challis on a new Equation in Hydrodynamics , 



serve to determine^ x and y as functions of the arbitrary 

 quantity %(^). These values, substituted in the equations 

 u=y{t — \)i v = x[t->r\)i show that the velocities are functions 

 of the same arbitrary quantity. Hence the proposed instance 

 is one of constrained motion, possible under given circum- 

 stances. 



The above interpretation will perhaps be made plainer by 

 taking a simpler example, het u = mWf v= —my, the fluid 

 being incompressible and no force acting. Then it follows 



from the equations -■=.—-■=. that the lines of motion 



V dy y 



are rectangular hyperbolas, and consequently, by mere geo- 

 metrical considerations, that the lines of displacement are 

 also rectangular hyperbolas of which the general equation is 

 oo^—y'^—a^ — O. Hence the equation \|/ = becomes 



Hence ^ =y'm ^ -2^ ^ _ _o„ a- ~ - - 



dx 

 Consequently the new equation becomes for this instance, 



x'W + 2w(a;2+y)=o. 



These two equations show that this is also an instance of con- 

 strained motion. I have remarked in the Cambridge Philo- 

 sophical Transactions, vol. viii. part 1 (Art. 3 of my paper on 

 a new Hydrodynamical Equation), that from the value of the 

 pressure p^ to which this example leads, viz. 



m 



p=if{t)-oi^^+y') 



2 



/J 



it follows, by putting p=0, that the boundary of the fluid is 

 at all times circular, a result incompatible with the hyper- 

 bolic lines of motion. This contradiction, I have asserted, 

 and still assert, proves the insufficiency of the recognized hy- 

 drodynamical equations. The contradiction is removed by 

 using the new equation. I think Professor Tardy would find 

 it difficult to show how, in the instance he has selected, sup- 

 posing W = 0, the boundary can at all times be a rectangular 

 hyperbola. 



Further to illustrate this point, let us take another instance. 

 Let 



«=^fWj '^=~f{^')i r^=a^+y^. 

 Then plainly si 



