attracted to ajixt Centre. 1 1 



pose R = 1 + tt, u being a small quantity ; then A = 1, and 

 we readily get, 



-~ COS 2 = ,— ri • 



For a first approximation, u = — cos 2 9, R = 1 + — cos 2 0; 

 and the fluid is an elliptical spheroid of revolution, the ellipti- 

 city being — , whereas, in the theory of Newton, it is 



5 



Carrying the approximation a step further, we have, 

 -?- cos 2 == u - 3u* : 



2 



and hence R = l+tt=l+-§- cos 2 + 3 -^- cos 4 9 ; 

 or, which is the same thing, 



R = 1 + (l+ 1 f)' ;osS »-TT sinS29 - 

 The figure of the fluid is no longer elliptical. Its surface falls 

 every where below an elliptical spheroid having the same polar 

 and equatorial diameters. The excess of the radius of the 



equator above the semi-polar axis, is increased from — to 



-i- + ^— , or it has received an addition of about ? i K of its 

 24 



former quantity. 



In general when the problem of the figure of the earth is 

 pushed so as to include quantities of the second order, the el- 

 lipticity will receive a variation which may amount to ^i^ of 

 the first approximation. But as the skill of the experimentalist 

 is insufficient, and in all probability will ever remain insuffi- 

 cient, to determine the earth's ellipticity nearer than ^th or 

 jq th part, it can be of no avail to be solicitous about a greater 

 degree of theoretical accuracy. The theory of Clairaut is pro- 

 portionate to the practical knowledge that can be acquired ; 

 to go further belongs to that species of philosophy which 

 strains at gnats and swallows camels. 



I remain, sir, &c. &c. 



July 11, 1826. James Ivory. 



III. Researches on the Theory of Hydro-dynamics. By Thomas 

 Tredgold, Esq. Civil Engineer*. 



TN this paper I propose to make a few remarks on the pro- 



-*■ positions which are commonly given as the basis of the 



theory of the resistance of fluids. It is well known that the 



* Communicated by the Author. 



B 2 results 



