extensive Class of Spheroids. 
6 1 
expression be taken, making a the only variable, we shall 
have 
where the expression on the right hand side is the attractive 
force parallel to a , as will readily appear by decomposing the 
direct attractions of all the molecules into the partial attrac- 
tions parallel to the co-ordinates. But, besides enabling us to 
find the attractive force in any proposed direction, the func- 
tion V has another advantage ; for it is this function, and not 
the expressions of the attractive forces, which enters into the 
equation of the surface of a body, wholly or partly fluid, in a 
state of equilibrium.* 
The expression for V, exhibited above is not of a commo- 
dious form, and on this account it becomes necessary to trans- 
form it. Let x — IT cos. S'; y — R' sin. S' cos. zr' ; and 2: = R/ 
sin. S' sin. zr ' ; then will R' be the line drawn from the mole- 
cule to the origin of the co-ordinates ; S' will be the angle 
which R' makes with the axis of x ; and zd the angle which 
the projection of R' upon the plane to which x is perpendicu- 
lar, makes with a line given by position in the same plane: 
from the assumed values of x,y, z, it is easy to derive these 
new values, viz. 
y ~ R' sin. S' cos. z>' == %/R ,z sin. ‘S'— 
z R' sin. S' sin. zr : 
and, by taking the fluxions so as to make x vary with R \y 
with S', and 2: with zs'\ which will leave dx, dy , dz, as well as 
dR\ dtf, dzr', unrelated and independent on one another as the 
* Mec. Cel. Liv. 3, No. 4. 
x = R' cos. S' 
= s/R'°— 
