ON BABBAGE’S ANALYTICAL ENGINE. 709 
pared for setting the mechanism in action, the upper indices of the 
Variables for data are all unity, and those for the Working and Re- 
_ sult-variables are all zero. In this state the diagram represents them*. 
¢ _ There are several advantages in having a set of indices of this nature ; 
but these advantages are perhaps hardly of a kind to be immediately 
perceived, unless by a mind somewhat accustomed to trace the success- 
__ ive steps by means of which the engine accomplishes its purposes. We 
have only space to mention ina general way, that the whole notation 
of the tables is made more consistent by these indices, for they are able 
_ to mark a difference in certain cases, where there would otherwise be 
an apparent identity confusing in its tendency. In such a case as 
_ Vn=V> + V,, there is more clearness and more consistency with the 
usual laws of algebraical notation, in being able to write ™+!Vn = Vp 
_ +”YV,, It is also obvious that the indices furnish a powerful means of 
__ tracing back the derivation of any result; and of registering various cir- 
cumstances concerning that series of successive substitutions, of which 
_ every result is in fact mere'y the final consequence; circumstances 
_ that may in certain cases involve relations which it is important to ob- 
serve, either for purely analytical reasons, or for practically adapting 
the workings of the engine to their occurrence. ‘The series of substi- 
_ tutions which lead to the equations of the diagram are as follow :— 
a)  @) (3.) 
1V,. %%V6—'V, 'Vo-!V,—'V5-1V, mn'’—m'n 
) (4.) 
Wie Von Ve Van Ve V5 1V, 4 ae mee 
Vi. 'Vs—'V, 'Vo.1V,—1V3.1V, mn’ —m'n 
_ There are three successive substitutions for each of these equations. 
_ The formule: (2.), (3.), and (4.) are implicitly contained in (1.), which 
_ latter we may consider as being in fact the condensed expression of any 
of the former. It will be observed that every succeeding substitution 
‘must contain ¢wice as many V’s as its predecessor. So that if a problem 
require 7 substitutions, the successive series of numbers for the V’s in 
the whole of them will be 2, 4, 8, 16...2”. 
_ The substitutions in the preceding equations happen to be of little 
_ value towards illustrating the power and uses of the upper indices; for 
_ owing to the nature of these particular equations the indices are all 
unity throughout. We wish we had space to enter more fully into the 
relations which these indices would in many cases enable us to trace. 
__ M. Menabrea incloses the three centre columns of his table under 
he general title Variable-cards. The V’s however in reality all repre- 
sent the actual Variable-columns of the engine, and not the cards that 
ng to them. Still the title is a very just one, since it is through 
special action of certain Variable-cards (when combined with the 
we generalised agency of the Operation-cards) that every one of the 
articular relations he has indicated under that title is brought about. 
ie Suppose we wish to ascertain how often any one quantity, or combi- 
_* We recommend the reader to trace the successive substitutions backwards 
from (1.) to (4.), in Mons. Menabrea’s Table. This he will easily do by means 
of the upper and lower indices, and it is interesting to observe how each V 
uccessively ramifies (so to speak) into two other V’s in some other column of 
the Table; until at length the V’s of the original data are arrived at. 
