[582 
TATION. 
;i874—1875), 
sssel’s Funda- 
mtre referred 
the Earth’s 
arth—I have 
s') 2 } + etc. 
582] NOTE ON THE THEORY OF PRECESSION AND NUTATION. 195 
The principal term is the first one, 
3 
- (xx' + yy' + zz'); 
but Bessel takes account also of the second term, 
- f p {O 2 + y- + z 1 ) (x' 2 + y 2 + z' 2 ) - 5 (xx' + yy + zz') 2 }, 
viz. considering the Earth as a solid of revolution (as to density as well as exterior 
form), he obtains in regard to it the following terms of sin co and ~ respectively; 
CLZ CLl 
or -I 
^ . 2 (C — A) K (5 sin 2 S — 1) cos 8 sin a, 
or 1 
~ 4r 4 Cn ‘ ^ ^ ~ ^ ^ ^ sin 2 ^ — ■*■) cos ^ cos a ’ 
where 
2 (C — A) K = S (3/a — 5/uL 3 ) 27rp R 5 dR dy, 
K being in fact a numerical quantity, relating to the Earth only, and the value of 
which is by pendulum observations ultimately found to be =0 , 13603. 
Writing, for shortness, 
(x 2 + y 2 + z 2 ) (x’ 2 + y' 2 + z' 2 ) — 5 (,xx' + yy' + zz') 2 = O, 
then the foregoing terms of sin co and depend, as regards their form, on the 
theorem that for any solid of revolution (about the axis of z) we have 
S(xy — xy') Qdm, S (y'z - yz') Qdm', S (z'x — zx') Qdm' 
= 0, 
\y (pc 2 + y 2 + z 2 — 5z 2 ) S [3 (x 2 + y' 2 + z' 2 ) — 5z' 2 ] z'dm', 
— % x (x 2 + y 2 + z 2 — 5z 2 ) S [3 (x' 2 + y' 2 + z' 2 ) — 5z' 2 \ zdm', 
respectively: viz. writing x' 2 + y' 2 + z' 2 — R 2 , and z' = Ryu, also x 2 + y 2 + z 2 — r 2 and 
x = r cos & cos a, y = r cos 8 sin a, z — r sin 8, the values would be 
0, 
\ r 3 cos 8 sin a. (1 — 5 sin 2 8) S (3 - 5y 2 ) yR 3 dm', 
— ^ r- 3 cos 8 sin a (1 — 5 sin 2 8) S (3 — 5y 2 ) yR 3 dm', 
which are of the form in question. 
The verification is easy: the solid being one of revolution about the axis of 2, 
any integral such as Sx'z' 2 dm' or Sx'y'z'dm' which contains an odd power of x' or of 
y' is = 0; while such integrals as Sx' 2 z'dm', Sy 2 z'dm are equal to each other, or, what 
is the same thing, each = \ S (x 2 + y' 2 ) z'dm'. That we have S (x'y - xy') Qdm' = 0 is 
25—2