Page:Scientific Memoirs, Vol. 3 (1843).djvu/724

714 {|
 * Variables for Results.
 * $$\scriptstyle{\mathbf{V}_{20}}$$
 * $$\scriptstyle{\mathbf{V}_{21}}$$
 * $$\scriptstyle{\mathbf{V}_{22}}$$
 * $$\scriptstyle{\mathbf{V}_{23}~\cdot~\cdot}$$
 * &ensp;
 * $$\scriptstyle{\cdot~\cdot~\mathbf{V}_{31}}$$
 * $$\scriptstyle{\mathbf{V}_{32}}$$
 * $$\scriptstyle{\mathbf{V}_{33}}$$
 * $$\scriptstyle{\mathbf{V}_{34}}$$
 * $$\scriptstyle{\cos\theta}$$
 * $$\scriptstyle{\cos2\theta}$$
 * $$\scriptstyle{\cos3\theta}$$
 * $$\scriptstyle{\cos\theta}$$
 * $$\scriptstyle{(\cos\theta.\cos\theta)}$$
 * $$\scriptstyle{\vdots}$$
 * $$\scriptstyle{(\cos3\theta.\cos\theta)}$$
 * $$\scriptstyle{(\cos2\theta.\cos\theta)}$$
 * }
 * $$\scriptstyle{\cos\theta}$$
 * $$\scriptstyle{\cos2\theta}$$
 * $$\scriptstyle{\cos3\theta}$$
 * $$\scriptstyle{\cos\theta}$$
 * $$\scriptstyle{(\cos\theta.\cos\theta)}$$
 * $$\scriptstyle{\vdots}$$
 * $$\scriptstyle{(\cos3\theta.\cos\theta)}$$
 * $$\scriptstyle{(\cos2\theta.\cos\theta)}$$
 * }
 * $$\scriptstyle{(\cos\theta.\cos\theta)}$$
 * $$\scriptstyle{\vdots}$$
 * $$\scriptstyle{(\cos3\theta.\cos\theta)}$$
 * $$\scriptstyle{(\cos2\theta.\cos\theta)}$$
 * }
 * $$\scriptstyle{(\cos2\theta.\cos\theta)}$$
 * }
 * }

The variable belonging to each coefficient is written below it, as we have done in the diagram, by way of memorandum. The only further reduction which is at first apparently possible in the preceding result, would be the addition of $$\scriptstyle{\mathbf{V}_{21}}$$ to $$\scriptstyle{\mathbf{V}_{31}}$$ (in which case $$\scriptstyle{B_1A}$$ should be effaced from $\scriptstyle{\mathbf{V}_{31}}$).|undefined The whole operations from the beginning would then be—

We do not enter into the same detail of every step of the processes as in the examples of Notes D. and G., thinking it unnecessary and tedious to do so. The reader will remember the meaning and use of the upper and lower indices, &amp;c., as before explained.

To proceed: we know that Consequently, a slight examination of the second line of (2.) will show that by making the proper substitutions, (2.) will become We shall perceive, if we inspect the particular arrangement of the results in (2.) on the Result-columns as represented in the diagram, that, in order to effect this transformation, each successive coefficient upon $\scriptstyle{\mathbf{V}_{32}}$,|undefined $\scriptstyle{\mathbf{V}_{33}}$,|undefined &amp;c. (beginning with $\scriptstyle{\mathbf{V}_{32}}$),|undefined must through means of proper cards be divided by two ; and that one of the halves thus