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It would stand as follows, previous to calculation:—

The data being given, we must now put into the engine the cards proper for directing the operations in the case of the particular function chosen.

These operations would in this instance be,—

First, six multiplications in order to get xn (=987 for the above particular data).

Secondly, one multiplication in order then to get a·xn (=5·987).

In all, seven multiplications to complete the whole process.

We may thus represent them:—

(×, ×, ×, ×, ×, ×, ×), or 7 (×).

The multiplications would, however, at successive stages in the solution of the problem, operate on pairs of numbers, derived from different columns.

In other words, the same operation would be performed on different subjects of operation.

And here again is an illustration of the remarks made in the preceding Note on the independent manner in which the engine directs its operations.

In determining the value of axn, the operations are homogeneous, but are distributed amongst different subjects of operation, at successive stages of the computation.

It is by means of certain punched cards, belonging to the Variables themselves, that the action of the operations is so distributed as to suit each particular function.

The Operation-cards merely determine the succession of operations in a general manner.

They in fact throw all that portion of the mechanism included in the mill into a series of different states, which we may call the adding state, or the multiplying state, &c. respectively.

In each of these states the mechanism is ready to act in the way peculiar to that state, on any pair of numbers which may be permitted to come within its sphere of action.

Only one of these operating states of the mill can exist at a time; and the nature of the mechanism is also such that only one pair of numbers can be received and acted on at a time.

Now, in order to secure that the mill shall receive a constant supply of the proper pairs of numbers in succession, and that it shall also rightly locate the result of an operation performed upon any pair, each Variable has cards of its own belonging to it.

It has, first, a class of cards whose business it is to allow the number on the Variable to pass into the mill, there to be operated upon.

These cards may be called the Supplying-cards.

They furnish the mill with its proper food.

Each Variable has, secondly, another class of cards, whose office it is to allow the Variable to receive a number from the mill.

These cards may be called the Receiving-cards.

They regulate the location of results, whether temporary or ultimate results.

The Variable-cards in general (including both the preceding classes) might, it appears to us, be even more appropriately designated the Distributive-cards, since it is through their means that the action of the operations, and the results of this action, are rightly distributed.

There are two varieties of the Supplying Variable-cards, respectively adapted for fulfilling two distinct subsidiary purposes: but as these modifications do not bear upon the present subject, we shall notice them in another place.

In the above case of axn, the Operation-cards merely order seven multiplications, that is, they order the mill to be in the multiplying state seven successive times (without any reference to the particular columns whose numbers are to be acted upon).

The proper Distributive Variable-cards step in at each successive multiplication, and cause the distributions requisite for the particular case.

The engine might be made to calculate all these in succession.

Having completed axn, the function xan might be written under the brackets instead of axn, and a new calculation commenced (the appropriate Operation and Variable-cards for the new function of course coming into play).

The results would then appear on V5.

So on for any number of different functions of the quantities a, n, x.

Each result might either permanently remain on its column during the succeeding calculations, so that when all the functions had been computed, their values would simultaneously exist on V4, V5, V6, &c.; or each result might (after being printed off, or used in any specified manner) be effaced, to make way for its successor.

The square under V4 ought, for the latter arrangement, to have the functions axn, xan, anx, &c. successively inscribed in it.

Let us now suppose that we have two expressions whose values have been computed by the engine independently of each other (each having its own group of columns for data and results).

Let them be axn, and bpy.

They would then stand as follows on the columns:—

We may now desire to combine together these two results, in any manner we please; in which case it would only be necessary to have an additional card or cards, which should order the requisite operations to be performed with the numbers on the two result-columns V4 and V8, and the result of these further operations to appear on a new column, V9.

Say that we wish to divide axn by bpy.

The numerical value of this division would then appear on the column V9, beneath which we have inscribed $\frac{ax^n}{bpy}$. The whole series of operations from the beginning would be as follows (n being = 7):

{7(×), 2(×), ÷}, or {9(×), ÷}.

This example is introduced merely to show that we may, if we please, retain separately and permanently any intermediate results (like axn, bpy) which occur in the course of processes having an ulterior and more complicated result as their chief and final object (like \frac{ax^n}{bpy}).

Any group of columns may be considered as representing a general function, until a special one has been implicitly impressed upon them through the introduction into the engine of the Operation and Variable-cards made out for a particular function.

Thus, in the preceding example, V1, V2, V3, V5, V6, V7 represent the general function Φ(a, n, b, p, x, y) until the function $\frac{ax^n}{bpy}$ has been determined on, and implicitly expressed by the placing of the right cards in the engine.

The actual working of the mechanism, as regulated by these cards, then explicitly developes the value of the function.

The inscription of a function under the brackets, and in the square under the result-column, in no way influences the processes or the results, and is merely a memorandum for the observer, to remind him of what is going on.

It is the Operation and the Variable-cards only which in reality determine the function.

Indeed it should be distinctly kept in mind, that the inscriptions within any of the squares are quite independent of the mechanism or workings of the engine, and are nothing but arbitrary memorandums placed there at pleasure to assist the spectator.

The further we analyse the manner in which such an engine performs its processes and attains its results, the more we perceive how distinctly it places in a true and just light the mutual relations and connexion of the various steps of mathematical analysis; how clearly it separates those things which are in reality distinct and independent, and unites those which are mutually dependent.

A. A. L.

Note C

Those who may desire to study the principles of the Jacquard-loom in the most effectual manner, viz.

that of practical observation, have only to step into the Adelaide Gallery or the Polytechnic Institution.

In each of these valuable repositories of scientific illustration, a weaver is constantly working at a Jacquard-loom, and is ready to give any information that may be desired as to the construction and modes of acting of his apparatus.

The volume on the manufacture of silk, in Lardner's Cyclopædia, contains a chapter on the Jacquard-loom, which may also be consulted with advantage.

The mode of application of the cards, as hitherto used in the art of weaving, was not found, however, to be sufficiently powerful for all the simplifications which it was desirable to attain in such varied and complicated processes as those required in order to fulfil the purposes of an Analytical Engine.

A method was devised of what was technically designated backing the cards in certain groups according to certain laws.

The object of this extension is to secure the possibility of bringing any particular card or set of cards into use any number of times successively in the solution of one problem.

Whether this power shall be taken advantage of or not, in each particular instance, will depend on the nature of the operations which the problem under consideration may require.

The process is alluded to by M. Menabrea, and it is a very important simplification.

It has been proposed to use it for the reciprocal benefit of that art, which, while it has itself no apparent connexion with the domains of abstract science, has yet proved so valuable to the latter, in suggesting the principles which, in their new and singular field of application, seem likely to place algebraical combinations not less completely within the province of mechanism, than are all those varied intricacies of which intersecting threads are susceptible.

By the introduction of the system of backing into the Jacquard-loom itself, patterns which should possess symmetry, and follow regular laws of any extent, might be woven by means of comparatively few cards.

Those who understand the mechanism of this loom will perceive that the above improvement is easily effected in practice, by causing the prism over which the train of pattern-cards is suspended to revolve backwards instead of forwards, at pleasure, under the requisite circumstances; until, by so doing, any particular card, or set of cards, that has done duty once, and passed on in the ordinary regular succession, is brought back to the position it occupied just before it was used the preceding time.

The prism then resumes its forward rotation, and thus brings the card or set of cards in question into play a second time.

This process may obviously be repeated any number of times.

A. A. L.

Note D

We have represented the solution of these two equations below, with every detail, in a diagram similar to those used in Note B; but additional explanations are requisite, partly in order to make this more complicated case perfectly clear, and partly for the comprehension of certain indications and notations not used in the preceding diagrams.

Those who may wish to understand Note G completely, are recommended to pay particular attention to the contents of the present Note, or they will not otherwise comprehend the similar notation and indications when applied to a much more complicated case.

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In all calculations, the columns of Variables used may be divided into three classes:—

1st.

Those on which the data are inscribed:

2ndly.

Those intended to receive the final results:

3rdly.

Those intended to receive such intermediate and temporary combinations of the primitive data as are not to be permanently retained, but are merely needed for working with, in order to attain the ultimate results.

Combinations of this kind might properly be called secondary data.

They are in fact so many successive stages towards the final result.

The columns which receive them are rightly named Working-Variables, for their office is in its nature purely subsidiary to other purposes.

They develope an intermediate and transient class of results, which unite the original data with the final results.

The Result-Variables sometimes partake of the nature of Working-Variables.

It frequently happens that a Variable destined to receive a final result is the recipient of one or more intermediate values successively, in the course of the processes.

Similarly, the Variables for data often become Working-Variables, or Result-Variables, or even both in succession.

It so happens, however, that in the case of the present equations the three sets of offices remain throughout perfectly separate and independent.

It will be observed, that in the squares below the Working-Variables nothing is inscribed.

Any one of these Variables is in many cases destined to pass through various values successively during the performance of a calculation (although in these particular equations no instance of this occurs) .

Consequently no one fixed symbol, or combination of symbols, should be considered as properly belonging to a merely Working-Variable; and as a general rule their squares are left blank.

Of course in this, as in all other cases where we mention a general rule, it is understood that many particular exceptions may be expedient.

In order that all the indications contained in the diagram may be completely understood, we shall now explain two or three points, not hitherto touched on.

When the value on any Variable is called into use, one of two consequences may be made to result.

Either the value may return to the Variable after it has been used, in which case it is ready for a second use if needed; or the Variable may be made zero.

(We are of course not considering a third case, of not unfrequent occurrence, in which the same Variable is destined to receive the result of the very operation which it has just supplied with a number.)

Now the ordinary rule is, that the value returns to the Variable; unless it has been foreseen that no use for that value can recur, in which case zero is substituted.

At the end of a calculation, therefore, every column ought as a general rule to be zero, excepting those for results.

Thus it will be seen by the diagram, that when m, the value on V0, is used for the second time by Operation 5, V0 becomes 0, since m is not again needed; that similarly, when (mn' − m'n), on V12, is used for the third time by Operation 11, V12 becomes zero, since (mn' − m'n) is not again needed.

In order to provide for the one or the other of the courses above indicated, there are two varieties of the Supplying Variable-cards.

One of these varieties has provisions which cause the number given off from any Variable to return to that Variable after doing its duty in the mill.