AND THE FIGURE OF THE EARTH. 
81 
Restoring the value of a, we get 
d*s , ds X — Y , s ,n 1 
dXdY + dX ntan 2 + ~2 , X - Y “ °» 
cos^-^- 
ds X — Y 
which is integrable ; and its integral is + n • « tan — = some arbitrary func- 
tion of Y, as % Y. Integrate again, then 
— f n tan Y~ A d Y ( P f wtan ^ ~ ^ dY \t i i 
S = S J 2 \J 2 *;Yfi?Y + ^Xj, 
where is arbitrary. Effecting these integrations, and reducing, 
-2.X-Y/ r 2ra X — Y A/JA7 . f 
S — COS — 2 — \J C0S — 2 — -^Y^Y + ^Xj. 
To return to P n , we have the following systems of equations : 
P % =fvdY, 
X- Y 
2 
d Y 
* = 
(/*/ tanL ir dY d Y) = cos' 2 ^/*i 
t cos ! 
Y-X 
cos 
- 4 
Y-X 
!/ 
q cos 4 
Y-X 
d Y, 
. Y, 
cos 
-2(»-i)Y-X 
/ 
.9 COS 
2(»-l)Y-X 
.dY, 
s= cos 2w -^^(ycos 2 ”^— ^ .^YrfY + ^x); 
whence 
P, = ..... _/cos 2 -/(/cos 
X , _ 2 
cos 
_ 2 Y — X 
(/os 2 * 
X- Y 
xYrfY + ^x)<TYdY.. .(w times.) 
Now cos ^T~ = cos (</ - 1 'osa/t^) = l(\/ 4=7 + V T T?)’ 
and cos 2 — g — = , _ ~ 2 ; and the complete integral will be expressed, by substituting 
for X and Y in terms of p and <p. 
6. But an important point yet remains to be determined. The original equation, 
being a partial differential equation of the second order, can only involve in its in- 
tegral two arbitrary functions. But here, after % Y and \p X have come in by two inte- 
grations, we have n integrations to perform with respect to Y. It would seem, there- 
fore, that no constant or arbitrary function of X must be added in these integrations. 
MDCCCXLI. 
M 
