the Attractions of Spheroids of every Description . 
21 
rational and integral function of f, V i — f 2 . cos. -sr', Vi -fiT 
. sin. Let x,y, % stand for f,s/i — f 1 . cos. -sr', 1 — 
. sin. ‘2r / ; and x',y f , z' for the analogous magnitudes y, V 1 — y* 
. COS. (p, s /7 — y . sin. (p: the first set of quantities are three 
rectangular co-ordinates of a point in the surface of the 
sphere whose radius is unit, drawn to the planes of three great 
circles two of which intersect in the origin of the arcs whose 
cosines are f and [a ; and the second set are the three rect- 
angular co-ordinates of the same point as before referred to 
three other planes two of which pass through the origin of 
the arc whose cosine is y : therefore, in order to obtain the 
relation of these two sets of quantities we have only to apply 
the method for transforming the co-ordinates : in this manner 
we shall readily obtain, 
x = x ' . p -f- y . s/ 1 — [/ 
y = x' . s / 1 — . cos. 7 zy - — y f . [a . cos. mr — %' . sin. tx 
z = x' . V i — f . sin. w — y . [a . sin. w + z' . cos. w. 
Because u' is a rational and integral function of x, y,z; by 
substituting the values of these quantities just investigated, it 
will be converted into a like function of x f , y', z, that is, of y, 
s/ i — y . cos. <p, s / 1 — y . sin. cp : and farther, if the several 
powers and products of cos. cp and sin. <p be exterminated by 
means of the equivalent expressions in the sines and cosines 
of the multiple arcs, the expression i/, after all the terms are 
properly arranged, will assume the following form, viz. 
u== r (o) + (i — fy, ( i — fy . r (l) . cos. cp + ( i — fy. 
(l — yf . r (z) . cos. 2 <p + &c. 
4- C 1 — pY • C 1 —/Y • ^ (l) • sin. <?> -f- (i—y) 1 . (i — y) 1 
. sin. 2 cp + &c. 
