514 
ON THE DEVELOPMENT OF THE DISTURBING 
[479 
Thirdly, 
D(j, —j + 4) = 2 
Hi (s + f) 
n (s + 2) 
n^(s- d)ni(s+ 6)+ 2 
II (s+ 2) 
x 
which, developed to rf, is 
~j + *) = v i 
-270*' i {SIH+ ° + 32H+ ‘>} : 
and, fourthly, 
D U, ~j + 6) = 2 V‘ 27,- (-)■ | 
II (s + 3) 
n* ( s — 0) (s + 6) + 3 
IT (s + 3) 
U(s + S) 
x 
n| (s + 6) II* (s — 6) + 3 
which, developed to rf, is simply 
D(j, -j + 6) = V 
The foregoing formulae, although obtained on the supposition j = 0, or positive, 
apply without alteration to the case j = negative, and the entire series of terms of an 
order not exceeding 6 as regards y may be written, 
D(j,-j) cos {jU-jU') 
+ 2D (j, —j + 2) cos (jU + (— j + 2) IT) 
+ 2D (j, —j + 4) cos (jU + (— j + 4) U') 
+ 2D (j, -j + 6) cos (jU + (-j + 6) U'), 
where j has every integer value from — oo to + oo. 
Comparison with Leverrier. 
This is in fact what Leverrier’s expression becomes on putting therein e = e' = 0. 
To verify this, observe that Leverrier having defined his A\ B\ O, D\ as above, writes 
further 
E l = \ (D i ~ 1 + B i+1 ), 
CD = | (O'- 2 + 40 + 0+ 2 ), 
H 1 = rpg (D { - 3 + 9D i+1 + 9D i+1 + D i+S ), 
D = | (O'- 2 + O), 
8 i = || (D 1 '- 3 + SD^ 1 + D i+1 ), 
T> = fa (J)i-3 + J)i-i) f