considered geometrically. 407 



dicular to the surface itself, which is the condition necessary for 

 equihbrium. 



The general formula dp-=pi^dx-\-Xdy-\-ldz) obviously be- 

 comes in this case 



dp=-p[--dx+-fdy+—dzj=-pAay-^+^-^ + -^J, 



and .-. » = C' — ^^(^ + 73 + ^-) at the surface jo = 0, and 

 ^ 2 \a- b^ c* / 



-5 + I2 +^ =lj •". C'=^^, and .-. the pressure at any point 



n \ n i V 11 2 \ 



a^y^-ofthe mass is =£-^(l ^ — j^ — -^),vi\ach.?Xi]\t centime, 



where x=y = z = 0, becomes ^^ ; and shows, moreover, that the 



surface of any ellipsoid concentric, similar, and similarly placed 

 to the given one, is a surface de niveau, on every part of which 

 the pressure is the same; and since A, B, C are obviously Gcp, 

 .". the pressure at any given point of the mass QC/3^. See the 

 Princifjia, Prop. 20, Book 3 ; M^Laurin's De Causa Phys. flux 

 et reflux Maris, Prop. 1 ; Airy's Tract on the Figure of the 

 Earth, Props. 14, 15 and 16. 



8. Let R and r be the radii of two homogeneous concentric 

 globes, A and a the attractions of each on a point on the surface 



A a 



of the other, then tts = -o-> whatever be the law of attraction as 



a function of the distance. 



For let be their common centre, OrR a radius meeting 

 them at r and R, be a chord of the less parallel to OR ; pro- 

 duce Oh, Oc to meet the large globe's surface at B, C, then 

 BC will be parallel to be or to OR; and if b describe any 

 httle figure b' on the surface of r, it is evident B will describe a 

 similar tigure B' on the surface of R ; and the areas S, s of the 

 normal sections of the cylinders C and c simultaneously described 

 by BC, Z»c- will obviously be to each other as B' : 6'; .*. as R^ : r^. 

 Now by Euclid (Prop. 4, Book 1) B?-=Z»R and Cr=cR ; .-. by 

 Airy's Tract on the Figure of the Earth, Prop. 18 (generalized), 

 attraction of cylinder C on the point r along ?-0 : attraction of 

 cylinder c on point R along RO : : S : s : : R^ : ?-^ ; and as this 

 fixed ])roportion holds true for each corresponding pair of cylin- 

 ders, .-. by taking their sums we shall still have A : a : : R^ : r^. 

 Sec Poisson's Mecanique, vol. i. p. 201. 



Mechanics' Institute, Liverpool, 

 AprU ly, 1854. 



