19 
the Attractions of Spheroids of every Description. 
For the greater simplicity we shall first consider the case 
when u' is a rational and integral function of p' only without 
w 1 , as is the case in spheroids of revolution. Suppose then o' 
— F (f): and by spherical trigonometry , 
f = py -j- Y 1 •— f . Y 1 — y m . COS. <p ; 
therefore by Taylor’s theorem, 
»'=F + 
+ ( 1 - / )* ( I - / )* . . cos.*<p+&c.* 
and by substituting for the powers of cos. cp their values in 
the multiple arcs, we shall have, 
t/ == r (o) -j- ( i — fy . ( i — fY . r (l) . cos. q> 
4- (i — yty . (l — y'Y . r (2) . cos. 2 (p -f- &c. 
the general term being ( i — fY . (l — yY x F 1 ^ . cos. i(p, 
where represents a rational and integral function of y. 
Now if we multiply by d(p, and then integrate between the 
limits <p = o and <p = 27 r, we shall get J ’ dd<p = 27r . r^ o) ; 
because the integrals of all the terms which contain the co- 
sines of the multiple arcs are evanescent at both the limits. 
Therefore, by substitution. 
JT 
(r — p) z 1 . p* . v . dp 1 . dm' 
2 7T 
/ 
(r-f> 
1 — 1 
. dy 
z 
| r z — zrp . y+p 1 j ill j r 2 ~ zr p . y + P 2 j 
In order to execute the remaining integration I remark that 
/ j r * — 24 . y -J- p a | % and dy=z-~- ; therefore by conti- 
, F (/xy) 
* By the notation — — — it is to be understood that in taking the fluxions, 
(py) is to be considered as one simple quantity; the same as if it were represented by 
a single letter. 
