Page:Scientific Memoirs, Vol. 3 (1843).djvu/739

for computing $\scriptstyle {B_{2n-1}}$, (in the case of $\scriptstyle {n=4}$); while the table beneath them presents a complete simultaneous view of all the successive changes which these columns then severally pass through in order to perform the computation. (The reader is referred to Note D, for explanations respecting the nature and notation of such tables.)

Six numerical data are in this case necessary for making the requisite combinations. These data are 1, 2, $\scriptstyle {n(=4)}$, $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$, $\scriptstyle {B_{5}}$. Were $\scriptstyle {n=5}$, the additional datum $\scriptstyle {B_{7}}$ would be needed. Were $\scriptstyle {n=6}$, the datum $\scriptstyle {B_{9}}$ would be needed; and so on. Thus the actual number of data needed will always be $\scriptstyle {n+2}$, for $\scriptstyle {n=n}$; and out of these $\scriptstyle {n+2}$ data, ($\scriptstyle {{\overline {n+2}}-3}$) of them are successive Numbers of Bernoulli. The reason why the Bernoulli Numbers used as data, are nevertheless placed on Result-columns in the diagram, is because they may properly be supposed to have been previously computed in succession by the engine itself; under which circumstances each $\scriptstyle {B}$ will appear as a result, previous to being used as a datum for computing the succeeding $\scriptstyle {B}$. Here then is an instance (of the kind alluded to in Note D.) of the same Variables filling more than one office in turn. It is true that if we consider our computation of $\scriptstyle {B_{7}}$ as a perfectly isolated calculation, we may conclude $\scriptstyle {B_{1}}$, $\scriptstyle {B_{3}}$, $\scriptstyle {B_{5}}$ to have been arbitrarily placed on the columns; and it would then perhaps be more consistent to put them on $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$ as data and not results. But we are not taking this view. On the contrary, we suppose the engine to be in the course of computing the Numbers to an indefinite extent, from the very beginning; and that we merely single out, by way of example, one amongst the successive but distinct series' of computations it is thus performing. Where the $\scriptstyle {B}$'s are fractional, it must be understood that they are computed and appear in the notation of decimal fractions. Indeed this is a circumstance that should be noticed with reference to all calculations. In any of the examples already given in the translation and in the Notes, some of the data, or of the temporary or permanent results, might be fractional, quite as probably as whole numbers. But the arrangements are so made, that the nature of the processes would be the same as for whole numbers.

In the above table and diagram we are not considering the signs of any of the $\scriptstyle {B}$'s, merely their numerical magnitude. The engine would bring out the sign for each of them correctly of course, but we cannot enter on every additional detail of this kind, as we might wish to do. The circles for the signs are therefore intentionally left blank in the diagram.

Operation-cards 1, 2, 3, 4, 5, 6 prepare $\scriptstyle {-{\frac {1}{2}}\cdot {\frac {2n-1}{2n+1}}}$. Thus, Card 1 multiplies two into $\scriptstyle {n}$, and the three Receiving Variable-cards belonging respectively to $\scriptstyle {\mathbf {V} _{4}}$, $\scriptstyle {\mathbf {V} _{5}}$, $\scriptstyle {\mathbf {V} _{6}}$, allow the result $\scriptstyle {2n}$ to be placed on each of these latter columns (this being a case in which a triple receipt of the result is needed for subsequent purposes); we see that the upper indices of the two Variables used, during Operation 1, remain unaltered.

We shall not go through the details of every operation singly, since the table and diagram sufficiently indicate them; we shall merely notice some few peculiar cases.

By Operation 6, a positive quantity is turned into a negative quantity, by simply subtracting the quantity from a column which has only