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LI. On some Theorems in Hydrodynamics. 

 By Professor Challis, M.A., F.R.S., F.R.A.S.* 



THE questions I now propose to consider are nearly identical 

 with those discussed in my communication to the Philo- 

 sophical Magazine for February 1853 under the head of Propo- 

 sitions XII. and XIII. As those propositions appear to me 

 absolutely necessary for completing the mathematical theory of 

 hydrodynamics, I have thought it right to confirm and elucidate 

 the reasoning by which they are established by additional argu- 

 ments. The course of reasoning here followed will be found to 

 be more explicit and satisfactory than that referred to, while it 

 leads to the same conclusions. 



It will be necessary for the present purpose to state briefly 

 the theorems that were demonstrated previous to the considera- 

 tion of Propositions XII. and XIII. Assuming as an axiom 

 that the lines of motion in each elementary portion of a mass of 

 fluid in motion are normals to a continuous surface, the equality 



7 • U 7 . V 7 ™ 1 



ohjr = - cte-f - ay + — dz 

 A, A A 



is true ; or, in other words, the function udx + vdy + wdz is in- 

 tegrable by a factor, without reference to any particular case of 

 motion. Now that condition of integrability is fulfilled in a 

 general manner if X be a function of yfr and t. After obtaining 

 (Prop. VI.), on the principle that the above axiom holds good 

 of any given element in successive instants, the general equation 



lt +x \d^ + df ^) ' 



it was shown (Prop. VII.) that if \ be a function of yjr and t, 

 that equation conducts to rectilinear motion. Next (Prop. VIII.) 

 a general equation was obtained on the two principles, that the 

 lines of motion in each element are normals to a continuous sur- 

 face, and that the mass of the element remains the same in suc- 

 cessive instants. If R, R' be the radii of curvature of the surface 

 of displacement, and V be the total velocity and p the density 

 at the time t, that equation is 



dpd.Vp^ v (\ .IV ft 



For an incompressible fluid, p being constant, the equation 

 becomes 



rfR +V \R + R7 u ' 



* Communicated by the Author. 



