The above expression (5) is general, and by no means complex. It may, however, be rendered more convenient for further deductions and corollaries by exterminating l and D'. Thus, for the first term,

sin D'tan (L — 1) = sin D cos I

tan L

tan /

1 + tan L tan /

sin Pмb

=

=

=

COS (L-1)

sin cos I cos L + sin 2 sin L

=

sin I

cos 2l (tan / cos L + tan sin L)

sin 1 (1 + tan 'L)

tan l cos L + tan 2/sin L

21

= [sin 1 (1 + tan)].

tan i cos D cos L + tan r sin L

sin I (sec - sin 3D)

tan I cos D cos L + tan 2 sin L

cos (sec 21 - sin 2D)
COS I COS D Cos L + sin I sin L

This is the general expression for the inclination of the hour line to the meridian: the first term is constant; the second has a constant coefficient to the variable

quantity cot P, which undergoing all its changes of magnitude in 90° = 6 x 15°, or 6 hours, the tracing of 6 hours of the dial will serve for tracing the rest. For the horary angle of the substyle we have

tan MS COS PMS tan PM = sin D' cot (L — 1), which, on comparison with the value of sin D′ tan (L —7) above obtained, evidently reduces to

sin I

sin l

+ sin L)

(cos L cos + sin L sin 7) = sin 1 (cos L cot l

the equation (4) becomes

sin E = sin I (sin L + cos L COS D cot 1) = sin I sin L + cos I COS D COS L.... (8.)

From these theorems (6), (7), (8), the whole of dialling on planes may be deduced.

17. Thus, let = 0, or the plane become erect; then from equa. (6), we have

By which the angle between the substyle and the hori

zon is determined.

And equa. (8) becomes

sin EcoS L COS D

....

(11.)

We have also, when м and z coincide,

cot zrs=sin L cot D.... (12.)

Thus we obtain the horary angle of the substyle.

18. If D as well as 1 become = 0, then

In this hypothesis also, we have

sin E = COS L . . . . (14.) . . . . . cot s = 0... . (15.) The three last formulæ evidently serve for erect direct south or north dials; in which equa. (15) shows that the substyle is perpendicular to the horizon. 19. If while I = O, D become = 90°;

then will equa. (10) be transformed to cot scot L (16), and equa. (11) to sin E = 0.... (17.)

This is the case of the erect direct east or west dial, in which it appears that the style is parallel to the plane, and the substyle inclined to the horizon in an angle equal to the latitude.

Here, since the style is parallel to the plane, the dial has no centre; all the hour lines, therefore, are parallel to the substylar line, which is the hour line of 6 o'clock: the respective distances of the hour lines from the substylar are measured by a cot P, on a perpendicular to

the 6 o'clock hour line; or by

acot p

COS L

on a vertical line.

This gives an infinite distance between the 6 o'clock and 12 o'clock hour lines; as there manifestly ought to be, because at noon the solar rays will be perpendicular to the plane of the dial.

20. If when D= O, we have 1 = 90°, the equations (6) and (8) become

or, tan мb = sin L tan P....

and sin E sin L (19.)

....

These theorems obviously apply to the horizontal dial. 21. If, in the last hypothesis, L become equal to 90°, we shall have

tan мb - tan P

sin E = radius.

This would apply to a horizontal dial at the poles, or a dial in any latitude with its face posited parallel to the equator. Here the formulæ show that the style would become a pin placed perpendicularly to the centre of

the dial, round which the hour lines would be radti drawn from the foot of the style to make angles of 15°, 30°, 45°, &c. with the meridian, or hour line of 12.

PROBLEM II.

22. To describe a horizontal dial for any proposed latitude.

This is the simplest dial to draw next to the polar dial, just mentioned; and it is the most useful, because, if it be posited where the sun's rays meet with no obstructions, that luminary will shine upon it from his rising to his setting. The theorems from which the construction is to be deduced are,

tan мb sin L tan P, and sin Esin L. se Here мb is the measure of the angle H, between the 12 o'clock and any other hour line on the dial, and P the hour angle from the meridian, as it varies at the poles of the heavens; the latter, therefore, varies uniformly, while the former only varies uniformly in certain limited cases; as, for example, in the horizontal dial at the latitude of 90°. In any other latitude, making the terms of the equation homogeneous, we have sin 90° tan H = sin L tan P.

Hence, if two radii be assumed in the ratio of sin 90° to sin L, tan H referred to the former radius will always be equal to tan L referred to the latter. From this consideration the following construction is deduced:

On the proposed

plane assume the right line 12 HS for the meridian, or 12 o'clock hour line, parallel to S which draw the line 12hs at a distance equal to the proposed. thickness of the style. Perpendicularly to 5 these draw 6H6, for

the east and west line of the dial, or the 6 o'clock hour line. Make the angle 12HF the latitude of the place, and from 12 let fall the perpendicular 12r upon HF. Make 12P, upon H12 prolonged, equal to 12r. From P draw lines P1, P2, P3, &c. (to terminate in the line 12-5, perpendicular to 12H) and to make angles 12p1, 12P2, 12P3, &c. equal to 15°, 30°, 45°, &c. Then from the centre H draw H1, H2, н3, н4, н5, for the hour lines of 1, 2, 3, 4, and 5, in the afternoon. Take, on the other side of the substylar line 12-11, 12-10, 12-9, &c. respectively equal to 12-1, 12-2, 12-3, &c. and from h draw the lines h7, h8, h9, &c. Produce the lines 4н, 5H, for the lines of 4 and 5 in the morning; and pro duce the lines 7h, 8h, for the hour lines of 7 and 8 o'clock in the afternoon.

The truth of this construction is manifest from the remarks which precede it. For 12P 12F is evidently the sine of the latitude to the radius H12. And while 12-1, 12-2, 12-3, &c. are the tangents of 15°, 30°, 45°, &c. respectively, to the radius P12; the same lines are tangents to the angles 1211, 12н2, 12н3, &c. Consequently, while the former are hour angles at the pole, the latter are the corresponding hour angles at the centre of the dial.

The quarter and half hour lines are drawn by setting off angles of 3° 45', 7° 30′, 11° 25′, &c. from the meridian line: but they are omitted in the diagram to prevent confusion.

The angle 12HF is also equal to the elevation of the style; for EL.

As to the gnomon, it should be a metallic triangle of the thickness hн, and having one angle L the latitude. It must be fixed perpendicularly to the plane, on the space left for it in the figure from hн towards 12, and having its angle L at hн: then will the style of the dial be parallel to the earth's axis.