Page:Scientific Memoirs, Vol. 3 (1843).djvu/719

pared for setting the mechanism in action, the upper indices of the Variables for data are all unity, and those for the Working and Result-variables are all zero. In this state the diagram represents them^[1].

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 successive steps by means of which the engine accomplishes its purposes. We have only space to mention in a 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 $\scriptstyle {\mathbf {V} _{n}=\mathbf {V} _{p}+\mathbf {V} _{n}}$ there is more clearness and more consistency with the usual laws of algebraical notation, in being able to write $\scriptstyle {^{m+1}\mathbf {V} _{n}=^{q}\mathbf {V} _{p}+^{m}\mathbf {V} _{n}}$. It is also obvious that the indices furnish a powerful means of tracing back the derivation of any result; and of registering various circumstances concerning that series of successive substitutions, of which every result is in fact merely the final consequence; circumstances that may in certain cases involve relations which it is important to observe, either for purely analytical reasons, or for practically adapting the workings of the engine to their occurrence. The series of substitutions which lead to the equations of the diagram are as follow:—

$\scriptstyle {{\overset {(1.)}{^{1}\mathbf {V} _{16}}}={\overset {(2.)}{\frac {^{1}\mathbf {V} _{14}}{^{1}\mathbf {V} _{12}}}}={\overset {(3.)}{\frac {^{1}\mathbf {V} _{10}-^{1}\mathbf {V} _{11}}{^{1}\mathbf {V} _{6}-^{1}\mathbf {V} _{7}}}}={\overset {(4.)}{\frac {^{1}\mathbf {V} _{0}.^{1}\mathbf {V} _{5}-^{1}\mathbf {V} _{2}.^{1}\mathbf {V} _{3}}{^{1}\mathbf {V} _{0}.^{1}\mathbf {V} _{4}-^{1}\mathbf {V} _{3}.^{1}\mathbf {V} _{1}}}}={\frac {d'm-dm'}{mn'-m'n}}}$
$\scriptstyle {{\overset {(1.)}{^{1}\mathbf {V} _{15}}}={\overset {(2.)}{\frac {^{1}\mathbf {V} _{13}}{^{1}\mathbf {V} _{12}}}}={\overset {(3.)}{\frac {^{1}\mathbf {V} _{8}-^{1}\mathbf {V} _{9}}{^{1}\mathbf {V} _{6}-^{1}\mathbf {V} _{7}}}}={\overset {(4.)}{\frac {^{1}\mathbf {V} _{2}.^{1}\mathbf {V} _{4}-^{1}\mathbf {V} _{5}.^{1}\mathbf {V} _{1}}{^{1}\mathbf {V} _{0}.^{1}\mathbf {V} _{4}-^{1}\mathbf {V} _{3}.^{1}\mathbf {V} _{1}}}}={\frac {dn'-d'n}{mn'-m'n}}}$.

There are three successive substitutions for each of these equations. The formulæ (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 twice as many $\scriptstyle {\mathbf {V} }$'s as its predecessor. So that if a problem require $\scriptstyle {n}$ substitutions, the successive series of numbers for the $\scriptstyle {\mathbf {V} }$'s in the whole of them will be 2, 4, 8, 16 … $\scriptstyle {2^{n}}$.

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 the general title Variable-cards. The $\scriptstyle {\mathbf {V} }$'s however in reality all represent the actual Variable-columns of the engine, and not the cards that belong to them. Still the title is a very just one, since it is through the special action of certain Variable-cards (when combined with the more generalised agency of the Operation-cards) that every one of the particular relations he has indicated under that title is brought about.

Suppose we wish to ascertain how often any one quantity, or combi-

1. 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 $\scriptstyle {\mathbf {V} }$ successively ramifies (so to speak) into two other $\scriptstyle {\mathbf {V} }$'s in some other column of the Table; until at length the $\scriptstyle {\mathbf {V} }$'s of the original data are arrived at.