Page:Scientific Memoirs, Vol. 3 (1843).djvu/725

obtained must be added to the coefficient on the Variable which precedes it by ten columns, and the other half to the coefficient on the Variable which precedes it by twelve columns; $\scriptstyle {\mathbf {V} _{32}}$, $\scriptstyle {\mathbf {V} _{33}}$, &c. themselves becoming zeros during the process.

This series of operations may be thus expressed:—

Fourth Series.

^[1]${\begin{aligned}&\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{32}\div 2+^{1}\mathbf {V} _{22}=^{2}V_{22}=BA_{2}+{\frac {1}{2}}B_{1}A_{1}}&\\&\scriptstyle {^{1}\mathbf {V} _{32}\div 2+^{1}\mathbf {V} _{20}=^{2}\mathbf {V} _{20}=BA+{\frac {1}{2}}B_{1}A_{1}.~.~.~.~.~.}&\scriptstyle {=C_{0}}\end{aligned}}\right.}\\&\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{33}\div 2+^{1}\mathbf {V} _{23}=^{2}\mathbf {V} _{23}=BA_{3}+{\frac {1}{2}}B_{1}A_{2}.~.~.~.~.~.}&\scriptstyle {=C_{3}}\\&\scriptstyle {^{1}\mathbf {V} _{33}\div 2+^{2}\mathbf {V} _{21}=^{3}\mathbf {V} _{21}=BA_{1}+B_{1}A+{\frac {1}{2}}B_{1}A_{2}}&\scriptstyle {=C_{1}}\end{aligned}}\right.}\\&\scriptstyle {\left\{{\begin{aligned}&\scriptstyle {^{1}\mathbf {V} _{34}\div 2+^{0}\mathbf {V} _{24}=^{1}\mathbf {V} _{24}={\frac {1}{2}}B_{1}A_{3}~.~.~.~.~.~.~.~.~.~.}&\scriptstyle {=C_{4}}\\&\scriptstyle {^{1}\mathbf {V} _{34}\div 2+^{2}\mathbf {V} _{22}=^{3}\mathbf {V} _{22}=BA_{2}+{\frac {1}{2}}B_{1}A_{1}+{\frac {1}{2}}B_{1}A_{3}}&\scriptstyle {=C_{2}.}\end{aligned}}\right.}\end{aligned}}$

The calculation of the coefficients $\scriptstyle {C_{0}}$, $\scriptstyle {C_{1}}$, &c. of (1.), would now be completed, and they would stand ranged in order on $\scriptstyle {\mathbf {V} _{20}}$, $\scriptstyle {\mathbf {V} _{21}}$, &c. It will be remarked, that from the moment the fourth series of operations is ordered, the Variables $\scriptstyle {\mathbf {V} _{31}}$, $\scriptstyle {\mathbf {V} _{32}}$, &c. cease to be Result-Variables, and become mere Working-Variables.

The substitution made by the engine of the processes in the second side of (3.) for those in the first side, is an excellent illustration of the manner in which we may arbitrarily order it to substitute any function, number, or process, at pleasure, for any other function, number or process, on the occurrence of a specified contingency.

We will now suppose that we desire to go a step further, and to obtain the numerical value of each complete term of the product (1.), that is of each coefficient and variable united, which for the $\scriptstyle {(n+1)}$th term would be $\scriptstyle {C_{n}.\cos n\theta }$.

We must for this purpose place the variables themselves on another set of columns, $\scriptstyle {\mathbf {V} _{41}}$, $\scriptstyle {\mathbf {V} _{42}}$, &c., and then order their successive multiplication by $\scriptstyle {\mathbf {V} _{21}}$, $\scriptstyle {\mathbf {V} _{22}}$, &c., each for each. There would thus be a final series of operations as follows:—

Fifth and Final Series of Operations.

${\begin{array}{c}\scriptstyle {^{2}\mathbf {V} _{20}\times ^{0}\mathbf {V} _{40}=^{1}\mathbf {V} _{40}}\\\scriptstyle {^{3}\mathbf {V} _{21}\times ^{0}\mathbf {V} _{41}=^{1}\mathbf {V} _{41}}\\\scriptstyle {^{3}\mathbf {V} _{22}\times ^{0}\mathbf {V} _{42}=^{1}\mathbf {V} _{42}}\\\scriptstyle {^{2}\mathbf {V} _{23}\times ^{0}\mathbf {V} _{43}=^{1}\mathbf {V} _{43}}\\\scriptstyle {^{1}\mathbf {V} _{24}\times ^{0}\mathbf {V} _{44}=^{1}\mathbf {V} _{44}}\end{array}}$

(N.B. that $\scriptstyle {\mathbf {V} _{40}}$ being intended to receive the coefficient on $\scriptstyle {\mathbf {V} _{20}}$ which has no variable, will only have $\scriptstyle {\cos 0\theta (=1)}$ inscribed on it, preparatory to commencing the fifth series of operations.)

From the moment that the fifth and final series of operations is ordered, the Variables $\scriptstyle {\mathbf {V} _{20}}$, $\scriptstyle {\mathbf {V} _{21}}$, &c. then in their turn cease to be Result-

1. It should be observed, that were the rest of the factor ($\scriptstyle {A+\Lambda \cos \theta +}$&c.) taken into account, instead of four terms only, $\scriptstyle {C_{3}}$ would have the additional term $\scriptstyle {{\frac {1}{2}}B_{1}A_{4}}$; and $\scriptstyle {C_{4}}$, the two additional terms, $\scriptstyle {BA_{4}}$, $\scriptstyle {{\frac {1}{2}}B_{1}A_{5}}$. This would indeed have been the case had even six terms been multiplied.