ON CERTAIN FORMS OF INTERPOLATION. 441 
INTERPOLATION TO Sixtus. 
(20"") Ot = blog, — gës Ant dis 4984. 
(157) dn d AA Vt... 
(95 : | —sg[4—saMTe.W...], 
siu ES 3. ate lO — sik + A s — Too) Hs], 
(117) d = rael — £j ...]. 
If 9, and #, are computed by (19), and 9, and 9, by using (15") and (9"), there 
will remain only 9, and 9; to be filled in. 
It will be observed that the series (8)... (12) and (15)... (17) always converge 
more rapidly than (20) and (21); and their use should sometimes be preferred on 
this account. A little reflection will often introduce, instead of the above-written 
fractions, some simpler equivalents, which may be mentally applied to the requisite 
degree of accuracy. Thus, in the examples given below, the: following have oc- 
curred : — 
Ye = i — wv 36 = + — 15v: 
so = ++ > to's = $ + ribs 
in all of which the right-hand term is easily applied when not small enough to be 
neglected. ! ; 
The function Y, in the examples given below, is the Moon's declination for 1865, 
 omitting the degrees; the values are taken where their higher differences are quite 
large. Two methods of interpolating to sixths (by halves and thirds, and then by 
thirds and halves) are given, and their results may be compared. For computing a lu- 
nar ephemeris, the first method here given has been found to possess peculiar advan- 
tages. An example of interpolating to fifths, i.e. to every tenth of a day, is then given 
on Encxe’s plan. The whole of the work which it is necessary to write down is printed. 
The columns headed S.8 6’, &c. contain the sums of each consecutive pair of values 
of the functions 8 O°, &c. Since the sum and difference of any two numbers must 
be both odd or both-even, in using (25) and (26) the nearest odd or even value has 
been taken for S. ER, according as 4 and z4f— ô were odd or even; and similarly 
with 9? and 9 in using (19) and (9”). In the interpolation to fifths, the computed 
third and second differences, 9% and J^, happen exactly to fill in the second and 
first respectively, all being carried to hundrédths of a second, i e. one place farther 
than the function Y. 
