ON BABBAGE’S ANALYTICAL ENGINE. 715 
obtained must be added to the coefficient on the Variable which pre- 
_ cedes it by ten columns, and the other half to the coefficient on the 
_ Variable which precedes it by twelve columns; V,., V; 3, &c. themselves 
becoming zeros during the process. 
This series of operations may be thus expressed :— 
Fourth Series. 
V5, = 2+ 1V,.=°V = BA, +2B, A; 
oe ae Sie bie io Aare Ade waa ae 
Wa, =-2+'1V.,="V.,= BA, +3B, Ag... = C,* 
. ee gee ett B,A+ 3B, A,=C, 
6 We 2 OV SV 2 Bye (id sts eaters vanes Cy 
; ee een te ee 
The calculation of the coefficients C,, C,, &e. of (1.), would now be 
completed, and they would stand ranged in order on Va, Vo,, &c. It 
will be remarked, that from the moment the fourth series of operations 
is ordered, the Variables V,,, V3., &c. cease to be Result-Variables, 
and become mere Working-Variables. 
__ The substitution made by the engine of the processes in the second 
_ side of (3.) for those in the first side, is an excellent illustration of the 
manner in which we may arbitrarily order it to substitute any function, 
number, or process, at pleasure, for any other function, number or 
process, on the occurrence of a specified contingency. 
We will now suppose that we desire to go a step further, and to ob- 
tain the numerical value of each complete term of the product (1.), that 
is of each coefficient and variable united, which for the (x + 1)th term 
would be C,, .cos 7 6. 
_ We must for this purpose place the variables themselves on another 
set of columns, V,,, V,., &c., and then order their successive multipli- 
ation by V.,, Veo, &c., each for each. There would thus be a final 
series of operations as follows :— 
Fifth and Final Series of Operations. 
Sey MON ge nN a 
SV X Vag = Vig 
8Vog X Vig = Vigo 
FVog X°Vis = Vis 
"Vox X Vag = Via 
_ (N.B. that V,, being intended to receive the coefficient on Vio which 
has no variable, will only have cos04(=1) inscribed on it, prepa- 
fatory to commencing the fifth series of operations.) 
_ rom the moment that the fifth and final series of operations is or- 
ered, the Variables V., V.,, &c. then in their turn cease to be Result- 
_* It should be observed, that were the rest of the factor (A+ A cos 6+ &c.) 
aken into account, instead of four terms only, C; would have the additional 
term 4B, A,; and C, the two additional terms, BA,,$B,A;. This would 
indeed have been the case had even siz terms been multiplied. 
