b26 V'HILOSOPHICAL TRANSACTIONS. [aNNO 1732. 



But these latter forces are opposite to the former. Therefore 



/, (_ M + ,) . {hb - Ibhr + rr) v" ' i ;„■" fk (b - hr) \ ■ 



^ ( ir--iir + -^ + -rr--} '' = ^• 



Hence ,s deduced \,^^).(,_,). + jt^^^ + r-" " ifnT) = 



_ a- (A — a) , j)a_ . Jah_ Jaa 



~~ »» + 1 ~^ n+\'^ b—a 2{b — a)' 



And in the cases when 7W =; — l, n=: — 1, there will be found, as above, 

 the equations of the sections, only changing the signs where they ought to be 

 changed. 



And by these radial equations may be found the equations to the co-ordi- 

 nates, as was done for the curve paq. 



And since the weight of the column, both in the superior and in the infe- 

 rior curve, ought to be the same, there v/ill be had an equation between the 

 weight A in the superior curve, and the weight a in the inferior; from which 

 there will be determined c«, to the ca before determined; and thus the whole 

 section of the fluid will be determined. 



Whatever the hypothesis of gravity be, the radius cd can be always obtained 

 of a given length, for any given angle dcp, and thus the figure of the fluid be 

 made either thicker or thinner, and that indeed in infinite ways; by putting 

 determinate values in the equation for li and r. Thus, the points p and q can 

 be made to coincide, by writing o for h and r; and then the section of the fluid 

 will consist of two oval figures joined at c. For there are an infinity of ratios 

 among tt, /j, andyj which will agree to that determination. 



For example, if the last be required, viz, that p and a may coincide in c, 

 there will result 



2 (?z + 1 ) Tri"- + ' = 2 (n + 1 ) ttC" + ' + 2 (m + 1 ) pac'" — IqfabC" - ' — qfaaC" - '. 

 Hence will arise infinite ratios among tt, p, and/. 



If we suppose the gravity bt)th towards y and towards c, to be in the simple 

 ratio of the distance from the centre; the section of the fluid will be a conic 

 section. And if it be then required that the points p, q, c, may coincide, the 

 figure will consist of two ellipses joined together at c. 



Now if the distance cy vanish, or the two centres coincide; then will i^ = o, 

 and e= a; and the fluid will form a spheroid. 



Further, if there be putm = n, and tt = o, the general equation for the sec- 

 tion of the fluid will become 2pr" + ' — {n+ \ )fa" ^ ' hhrr = (2/)— nf—f)a'' + > . 



Or in the case of n = — I, it is 2pa log. - ='— Ja, as we found in the 



first problem, which is only a special case of this. 



Scholium. — This consideration of the figures which fluids may take, accord- 



