Page:Climatic Cycles and Tree-Growth - 1919.djvu/113

Now, taking analyzing lines aa¹ and bb¹ in figure 31 as horizontal, and letting the sweep be inclined as a small angle δ with the analyzing lines, the number of lines required to cross the sweep in the direction ab perpendicular to analyzing lines will be increased and hence the value in years between two analyzing lines will be decreased; hence

$\scriptstyle {ys \over l}cos\delta$ = years per line from a to b.

If the fringe is perpendicular to the analyzing lines, its period is the distance ab in years and we have for this special case:

$\scriptstyle p_{1}={ys \over l}cos\delta$.

Fig. 31.—Diagram of theory of differential pattern in periodograph analysis.

If, however, the fringe takes some other slant, as the direction ac, making the angle θ with the analyzing lines, then the period desired is the time in years between a and c. That equals the time between a and b less the time from b to c. Now bc in years would equal $\scriptstyle {\overline {ab}}cot\theta$ except for the fact that the horizontal scale along bc is greater than the vertical scale along ab in the ratio $\scriptstyle {cos\delta } \over {sin\delta }$ and therefore a definite space interval along it means fewer years in the ratio of $\scriptstyle {sin\delta } \over {cos\delta }$. Hence we have:

bc (in years) = ab (in years) tan δ cot θ

or

$\scriptstyle P=p_{1}\left(1-\tan \delta \cot \theta \right)$

which is the period required.

The separation of the fringes needs to be known at times in order to find whether one or more actual cycles are appearing in the period under test. In figure 31

$\scriptstyle {\overline {ab}}=s$

$\scriptstyle {\overline {ad}}={s \over {sin\delta }}$

$\scriptstyle {\overline {ac}}={{ssin(\theta -\delta )} \over {sin\delta }}$

which is the width required.