HOMOGENEOUS FLUID AT LIBERTY. 529 



passing from one to another, is perfectly general ; it embraces the full extent of the 

 problem, and comprehends the first mode of action as a particular case. It happens 

 that either of the two ways of rendering the attraction of a stratum ineffective to 

 move the particles contained within it, leads precisely to the same final results in 

 determining the figure of equilibrium of a homogeneous planet, which, although it 

 does not excuse the misconception, makes the correcting of it less diflScult. In con- 

 clusion, what is exceptionable in the paper of 1824 has already been explained pub- 

 licly; and the paper in the Philosophical Transactions for 1831 is not liable to the 

 same reproach. 



Appendix, containing the Investigation o^ some Algebraic Formulas. 



1. Development of / ^ ^ „ , used in No. 7- 



r y^2 — 2 sry + r^ 



If we assume 



s — rz = ^s" — 2 sry -\- r^y 

 the value of z will be 



1 r 



^ = 7 + -Q- • 7" (^^ - 1) •• 



now considering 2 as a function of y, and applying the theorem of Lagrange/ we 

 deduce 



and by substituting this value in the assumed formula, we obtain 



Js^ -2*ry + r2 = * — ry— "2 . y (y2 _ 1) - y;^ . -^ . ^'—^ ^c. ; 



then by differentiating with respect to y, and dividing by s r, we finally obtain 



K^s^-^^sry + t^'^ 5 "^ 2 • s^ • dy ^ I . 2 ' 2^ * s^ ' dy -r,<^^' 



This expression of the development is investigated differently in the Philosophical 

 Transactions for 1824. 



2. Demonstration of the theorem used in No. 7. 



It is obvious that 6 and & are the two sides of a spherical triangle, tsr — nx' being 

 the included angle, -4/ the third side, and <r the angle opposite to & ; wherefore, be- 

 cause y = cos -^1/ and ^1 — y^ = sin -v//, we have by the known properties of spherical 

 triangles, 



cos 6' = cos ^ . y 4- sin ^ . /v/T— y^ cos a 



sin & sin {vy — vx') = ^1 — y^ sin c 



sin & cos {rsT ■— nr') = sin ^ . y — cos d . ^1 —■ y^ cos c. 



