512 
ON THE DEVELOPMENT OF THE DISTURBING 
[479 
is taken to be developed in multiple cosines of U, U', the general term being 
B(j, j') cos {jU+j'U'), 
where j, j’ have each of them any integer value from — co to + x (zero not excluded), but 
so that j,j' are simultaneously even or simultaneously odd. We have D (—j, — j') = D(j,j') 
and D (f, j) = D(j, j'); and it hence appears that the really distinct values of the 
coefficient may be taken to be those for which j is not negative, and as regards 
absolute magnitude is not less than j'; and for such values of j, j' we have the above- 
mentioned expression 
D (j, f) = 2 rT TlMS RJ-, 
which I proceed to explain and develope. 
U, (x — |-) and Tlx (x being a positive integer) denote respectively ^. | ... (x — \), 
and 1.2.3... a?; in particular for x=0, the value of each factorial is =1. 
7] denotes sin ^ <£. 
The coefficients R x % are those of the multiple cosines in certain developments, viz. 
we have 
i a, r' x [r 2 + r 2 — 2rr' cos(U — U')}~ x ~- = %RJcosi(TJ — U'), 
where, as usual, i extends from — x to x and R x ~ l = RJ. Writing with Leverrier 
(a 2 + a 2 — 2aa' cos H)~^ = ^ cos iH, 
aa' (a 2 + a 2 — 2aa! cos If) = i ^B l cos iH, 
a?a' 2 (a 2 + a 2 — 2aa cos H)-% = SO’ cos iH, 
a 3 a' 3 (a 2 + a' 2 — 2aa / cos H) ~ * = ^ ID 1 cos iH, 
then 2Rj, 2R 1 i , 2Rj, 2Rj are the same functions of r, r that A\ B i , C\ D l respectively 
are of a, a'. 
The expression of M x % is 
M’» = ti+fl . 
v (*+/ + ») ni^-j + ^ni^+z-a)’ 
and, finally, in the expression for D (j, j'), x has every integer value from 0 to x, 
and, for any given value of x, ^ extends by steps of two units from the inferior 
value -{x -/) to the superior value x—j. 
It is convenient to write x = \ (j +f) + s; we have then ^ extending from 
“2 (j-j')~ s or writing S-=—Hj“/) + #> 0 has the s + 1 values s, 
5-2, s 4, ... s, viz. for s = 2p+l the values are ±1, ± 3,... + (2p +1), and for 
s=2p they are 0, ±2, + 4,... + 2p.