1816.) Scientific Intelligence. 477 
a. _ @y d*y w? 
. a ee +a + Te Fhe 
dy 
72 dy d*y 2? we? 
% sf — _ ase 
€ 1 jc + te is &e 
y 
2 3w dy d? yy 37%? . 
iy 1 = 7° +75 . Tra. + &e. 
@eeeesveeeeeeeeaeeee seene . essere rvreoeeeeeoeos 
Sew dy d® yw? 
= — —_ 
¢ Be RO ach greater he 
provided in the developements of the first members we transfer the 
exponents of the powers of d y to the characteristic d. 
Hence we may obtain 
Tz £19 dy wo? 
y+ NB Ae He w+ oe Te t &. 
—e d da? 2 
y+ e* Sm lsyt Powe sd. FS 4 &e, 
yr Or payee we Ee, Se, Be, 
1.2 
ay mw 
aa * 1.2 
And since the second members of these equations are identical with 
the corresponding members of the equations represented by (A), 
we shall obtain 
+ &e. 
y =yY 
£8 wy 
¥, = y + &* Tid 
d 
Y= yt er) 
Ye, = yt &* Pe 
diinany tenia tigks dau 
Yn = y -- et ath, i] 
But by the theory of finite differences 
A YFPMe Mahl 
A* YHR¥e7 2H + y 
=, ~3Y, + 3y, + y 
A" Y= Yn ~ 1 Ynnr + 7 Ynoe — Be, 
And substituting the last found values of Hee bs 
we shall prin en Yio Yoo Vs Yn» 
