AND THE FIGURE OF THE EARTH. 
83 
Y — X 
tan — ^ — , whose coefficients may, for anything we yet know, be functions of X. 
The following process shows that constants as coefficients will satisfy the original 
equation, and determines them. The integration itself gives the coefficient of the 
highest power. 
Let then* 
( ~2 n ~ 1 „ Y — X , „ ,Y-X .. „ 2 Y-X-| 
tan — ^ b d tan” ~ 1 — 5 b c tan” ~ 2 — ■= — 1 
P = c< 
[ri] 2 
+ • • • + c (n ~ 2) tan 2 + c (w - 1} tan + c (n) 
Then 
r 2 n ■ 
d P 
d Y 
? = c ' 
tan «-.I^ + ^Vtan--^ Y -X 
[» — 1] 2 "T" 2 
w 2 1/ — 3 Y X 
+ 
c" tan" 
+ ••••+ 2 c 
a ( »_2) + Y-X 
— A ’ tan — s — 
. _L c ( »-u 
^ 2 c 
O + tan*^*), 
f 2 W 
= ci 
[»— 1] 
tan 
Y - X , n - 1 . „ Y — X . (n — 2 
2 
+ (^ 3 c”' + ’^c') 
tan” - 
+ 
c tan 
+ C " + 
^ V 2 ^ [»_ !]/ 
tan 
n — 1 
y - x 
+ C 
n — 4 
tan 
« — 2 
Y-X 
c h -f 
n — 2 
Y-X 
-f ... + (I c ( ” “ 3) + c (w-5) ) tan 4 
Y-X 
(§- c^* “ 2) + | c<" - 4 ) ) tan 3 
Y-X 
+ (y c< w - + | c<* - 3 >) tan 2 ^y-^ - 2 ) tan + {c^ - ^ 
r (» + l)g»T» w Y-X , » (n + 1 ) ,^_i Y-X 
[» - 1 ] 
tan” ^ _J_ c r tan » 
d* P, 
dXdY 
cos 
Y-X 
— 1 (n — 2 
+ *-^( 
n — 2 (n — 3 
C 
9 rc — z \ 
" + 
w — 2 
tan 
n — 3 
Y-X 
2 
Y-X 
Y - X 
2 
+ 2 
+ • . . -b y (y c ( ” ~ 3) + “ 5) ) tan 3 
+ I(|^- 2) + 4c ( ”- 4) ) tan 2 
+ T(y c(K_1) + T c(K “ 3) ) tan^-=-^- <*» “ 2 > 
2 2 
* [»] = 1 . 2 . 3 . 4 . . . . n. 
M 2 
