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LIX. On ike Necessity of Three Fundamental Equations for 

 the general anali/tical determination of the Motion of Fluids, 

 By the Rev. J. Challis, M.A., Plumian Professor of As- 

 tronomy and Experimental Philosophy in the University of 

 Cambridge'^. 

 'l^HE communications which I have made from time to time 

 -*■ to this Journal on the analytical theory of hydrodyna- 

 mics have had particular reference to the mathematical jprin^ 

 ciples of this department of science. In the course of my in- 

 quiries I have been led to the conclusion, that those principles, 

 as commonly received, are essentially defective. I have pointed 

 out wherein I conceive them to be defective, and what appears 

 to be necessary to complete them. These views have not been 

 refuted, nor, as far as I am aware, have they been assented to. 

 I propose, therefore, in this communication to collect together, 

 in as brief a compass as possible, the chief arguments I have 

 employed, and to add others in support of them, for the pur- 

 pose, more especially, of substantiating the main result I have 

 arrived at, on which, in fact, all that I have advanced at vari- 

 ance with the views of other writers depends, viz. the necessity 

 of a third general equation in hydrodynamics. 



It will, I suppose, be granted, that in whatever manner a 

 mass of fluid is in motion, an unlimited number of surfaces 

 may be conceived to be drawn at each instant, cutting the di- 

 rections of motion at right angles. Let two such surfaces be 

 drawn at a given instant indefinitely near each other, and let 

 one pass through a point P given in position. On this sur- 

 face describe an indefinitely small rectangular area having P 

 at its centre, and having its sides in planes of greatest and 

 least curvature, and let normals be drawn at the four angular 

 points. By a known property of curve surfaces these normals 

 will meet, two and two, in the two focal lines situated in the 

 planes of greatest and least curvature. Let the small area, of 

 which P is the centre, be m^, and at a given time, t, let r, r' 

 be the distances of the focal lines from P, or the principal 

 radii of curvature. Produce the normals to meet the other 

 surface in four points, and join these points so as to form an 

 area on the second surface corresponding to the area ;»^ on 

 the first. Then, if the two areas be supposed to be ultimately 

 parallel, and to be separated by a given small interval S r, the 



second area is m^ . - — — —-^ , at the given time. As in 



general the direction of motion through the fixed point P 



varies continually, the normal surface through that point will 



also vary its position with the time. The positions of the 



* Communicated by the Author. 



