lOl 
in a series of the attraction of a spheroid. 
all the functional quantities being integral expressions of jx. 
The general term of this expression is 
{ F ( to + Vi— v ?f%) } cos i<p 
-f- { G %) + “V 7 1— pAg %) } sin i(p ; 
t 
and by multiplying by or 1, it will become. 
f f ( V) . / ( V) 1 1 
1 T -t E? f( 1 — t* ) 
1(1 — ^) a ( I— At 2 ) 2 J 
.(0 
COS Kp 
+ 
Ji) 
sini> 
l(l~f* 2 )3 (l-^) 2 J 
and finally, by expanding the denominators, we get, 
M w (t— p . 3 ) 2 cos itp -f- N w (l— -(x 2 ) 2 sin i(p; 
(0 
(«) 
the symbols and denoting rational series of the 
powers of p,. By performing these operations in all the 
terms containing cp, and likewise by expanding the radical 
V i — p in the first term, the value of y will be thus ex- 
pressed ; 
1 2 
y=M ( 0 ) 4 -M Cl} ( | — [x *) 2 coscp-{-M (2) (l — {x 2 ) 2 cos 2 <p &c. 
x 
+ N (1) (i — | it 2 ) 2 sin <p+N (2) (x — [x 2 ^ 2 sin 2 <p 
all the functional symbols standing for series of the powers 
of (x. The given expression now consists entirely of combi- 
nations of the quantities p,, f . cos <p t V 1 — jx 2 . sin <p; 
that is, it is a function of three rectangular co-ordinates. 
The same end might have been accomplished by a shorter 
and more simple process, which will apply to every function 
of two variable arcs. 
