Page:Scientific Memoirs, Vol. 3 (1843).djvu/743

means of a subtraction; and so on; but we shall not enumerate further. It is desirable in all calculations, so to arrange the processes, that the offices performed by the Variables may be as uniform and fixed as possible.

Supposing that it was desired not only to tabulate $\scriptstyle {\mathbf {B} _{1}}$, $\scriptstyle {\mathbf {B} _{3}}$, &c., but $\scriptstyle {\mathbf {A} _{0}}$, $\scriptstyle {\mathbf {A} _{1}}$, &c.; we have only then to appoint another series of Variables, $\scriptstyle {\mathbf {V} _{41}}$, $\scriptstyle {\mathbf {V} _{42}}$, &c., for receiving these latter results as they are successively produced upon $\scriptstyle {\mathbf {V} _{11}}$. Or again, we may, instead of this, or in addition to this second series of results, wish to tabulate the value of each successive total term of the series (8), viz: $\scriptstyle {\mathbf {A} _{0}}$, $\scriptstyle {\mathbf {A} _{1}\mathbf {B} _{1}}$, $\scriptstyle {\mathbf {A} _{3}\mathbf {B} _{3}}$, &c. We have then merely to multiply each $\scriptstyle {\mathbf {B} }$ with each corresponding $\scriptstyle {\mathbf {A} }$, as produced; and to place these successive products on Result-columns appointed for the purpose.

The formula (8.) is interesting in another point of view. It is one particular case of the general Integral of the following Equation of Mixed Differences:—

$\scriptstyle {{\frac {d^{2}}{dx^{2}}}\left(z_{n+1}x^{2n+2}\right)=(2n+1)(2n+2)z^{n}x^{2n}}$

for certain special suppositions respecting $\scriptstyle {z}$, $\scriptstyle {x}$ and $\scriptstyle {n}$. The general integral itself is of the form,

$\scriptstyle {z_{n}=f(n)\cdot x+f_{1}(n)+f_{2}(n)\cdot x^{-1}+f_{3}(n)\cdot x^{-3}+\cdots }$

and it is worthy of remark, that the engine might (in a manner more or less similar to the preceding) calculate the value of this formula upon most other hypotheses for the functions in the integral, with as much, or (in many cases) with more, ease than it can formula (8.).

A. L. L.

 

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