724

This value (X cos. 9) of c corresponding to the middle of the *eclipse, is the least distance, or, the nearest approach of thecentres of the Moon and shadow. For, if by the rules for findingthe maxima and minima of quantities, we deduce from the expres-sion, p. 723, 1. 3, the value of t, it will be found equal toX sin.* 9n

The nearest approach of the centres being known, the magni'tude of the eclipse is easily ascertained. Thus, on the sup'

, d D\

position that X cos, 9 is less than the distance ^P -f p + - —

at which the Moon ’s limb just touches the shadow, some part ofthe Moon ’s disk is eclipsed ; and the portion of the diameter ofthe eclipsed part is

P + P + - -X cos. 9.

2 2

The portion of the diameter of the non-eclipsed part, is d* eMoon ’s apparent diameter (d) minus the preceding expression;and, therefore, is

d D

' ' X cos. 9 - J— -1-P — p.

2 2 r

If this expression should be equal nothing, the eclipse wouldbe just a total one. If the expression should be negative, th eeclipse may be said to be more than a total one, since the upp e1 ’boundary of the Moon ’s disk would be below the upper boundaryof the section of the shadow : and the distance of the two boun-daries would be the preceding expression.

The preceding formulae for the parts eclipsed, which are p artsof the Moon ’s diameter, are usually expressed in twelfths of th®*diameter ; which twelfths are, with no great propriety of languagjhcalled Digits. Thus, if the part eclipsed should be 24/ 52 >the Moon ’s diameter being 33 ; 18 w ; then, the part eclips^24' 5<2," DijitJ. Digits.

= 5TT? * 12 = 8 - 96 -

By p. 723, the second root of the quadratic, or