CONGRUENCE 107

whence, if k 1s odd, 7y = Sx@ — b
and, if k is even, 7, = — Spa + b,
or ska — b = (— 1)¥ g,
whence, if k 1s odd, ska — b = 13
if kis even, sx@ — b = — 1p

Set I, = (— 1)¥ s and m; = (— 1)*; for all values of %

from — 1 to = inclusiv.

Then, remembering that any odd power of — 1 equals — 1

and any even power equals 1,

l_l = (— 1)_23_1 = m_, = (— 1)_1t_1 =it
Iy, = (=17 =0 me =(—1)% =1
Also,if 0 <k <n -+ 1,
hiea — Qulp— = (— 1)¥3s_9 — gi(— 1)*¥2s5;_4
= (= 1)* 53 + aal— 1) 2514
= (— 1D*Ys1—s + QkSk—l)
= (— 1)k g
= lk
Hence, lf 0 < E<mn + 1, lk = lk_g e lek—l
Similarly, if 0 < & < n + 1, mp = mr—s — QMr—
Hence I;, and m; are the same as in § 239.
Now (— l)k—l(ska = tkb) = (—' l)k‘lska + (— 1)ktkb
= lka'—l— mkb
=i § 239.

Therfor 7, = (— 1)*(sza — #b) for all values of k2 from

— 1 to » inclusiv.

This theorem can also be proved by mathematical induction

in the same way as § 239.

and tx

255. The formulas Sk = Sk—2 + QrSk—1
= lp—2 + Qrlr—

show how, having given s_i, So, -1, fo, and the series of ¢’s,
we may find all the s’s and #’s.

The work is like that of § 241, but where in that article we

subtracted, here we add.