ON BABBAGE’S ANALYTICAL ENGINE. 725 
which is in fact a particular case of the development of 
a+bex+ca?+4+ &e. 
a+b'a+c'a2+ &e. 
mentioned in Note E. Or again, we might compute them from the 
well-known form 
bs 2iS s058n 1 i } 
Bye a eee Pees «te ; 
an—1= 2 (aay {i++ 4 \ (2.) 
g2n 
or from the form 
F 1 2n-1 
2 
2n—1 1 2n 
(may fg 
2n-1 2n,1 2n.(2n—1 
col +2n +(n—2) {i+ a7 } (3.) 
ene —s)a" Ane2n 1s 
2n-1 2 
wt amid pate 1.2 
1 2n.(2n—1).(2n—2) 
L +z" 1.2.3 
Ute 
or from many others. As however our object is not simplicity or facility 
of computation, but the illustration of the powers of the engine, we pre- 
fer selecting the formula below, marked (8.). This is derived in the 
following manner :— 
If in the equation 
+B 
B “fh Hie ie ily 
=1—— +B,—+B, —~— ae eee reWes 
a | lv dnetae RRC ROR °2.3.4.5.6 
(in which B,, B,;...., &¢. are the Numbers of Bernoulli), we expand 
the denominator of the first side in powers of z, and then divide both 
numerator and denominator by 2, we shall derive 
7) a? at ey ae x3 
= — — Bie | pas eee _ — eee 
1=(1-5+B,5+B55 4+ )QG+§tegtaga-)@ 
If this latter multiplication be actually performed, we shall have 
a series of the general form 
(wa We ae ee be eae ow 1%) 
in which we see, first, that all the coefficients of the powers of z are 
severally equal to zero; and secondly, that the general form for Ds » 
the co-efficient of the 2x + 1th ¢erm, (that is of 2°” any even power of 
2), is the following :— 
paves) 
ie ak ct Mey, Bis GoMod tt BE 1 
9.3...9n+1 22.3..29n' 2°2.3..9n—1' 2.3.4 a 
enlied tees ret el Oe | 
2.3.4.5.6 2.3...2n—5 2.3...2n J 
