perpendicular to the plane of the horizon, are called inclining or reclining dials. They may, at the same time, be either direct or declining, according as they present a sloping face to the cardinal points, or not.

10. The arch of the horizon which is intercepted between any given plane, and that of the prime vertical, is called the declination of the plane. It will be denoted by D, and will be regarded as positive when the declination is towards the north, negative when it is towards the south.

11. The inclination, 1, of a plane is the angle which it makes with a vertical plane.

12. The intersection of the plane of the dial and that of the meridian passing through the style is called the meridian of the dial, or the hour line of 12.

13. Those meridians passing through the style, which make angles of 15°, 30°, 45°, &c. with the meridian of the place (marking the hour line of 12) are called hour circles, and their intersections with the plane of the dial hour lines.

14. The angle formed by the substyle and the meridian is called the horary angle of the substyle: it will be denoted by M.

15. The angle included between the substyle and the horizon is termed the inclination of the substyle: it will be denoted by s.

GENERAL PROBLEM.

16. To determine the requisites in a dial of any proposed inclination to a vertical plane, and declination from the prime vertical.

Let HOR, in the marginal diagram, represent the horizon, HZR the meridian of the place, z the zenith, zo the prime vertical, P the elevated pole, H Pb, Po, portions of hour circles, zeb, zrb', corre

M

Ꮓ

sponding hour angles, and let the plane on which it is proposed to draw the dial be coincident with the plane of the great circle Moт. Also, let za perpendicular to MaT be drawn and produced to 1. Then PRL, the latitude of the place; za 1, the inclination of the plane; OT = HI = D, its declination; Ps drawn perpendicular to MT will be the position of the substyle; and mb is the inclination of the hour line to the meridian.

Now, we may regard the dial whose plane coincides with Mat, as a vertical or erect dial at the place whose zenith is м, where м and z are on the same meridian, and, of course, reckon the hours alike. Let мz = 1, and D' comp. of zмɑ. Then PZ 90° L would become PMPZ + ZM = 90° - 1 + 1 = 90° — (L—7). The right angled triangle мza, gives

tan Masin za tan мza = sin 1 tan D . . . . (1.) Here мa is the angle between the meridian and vertical, and is obviously evanescent when either I or D are. When I = 90°, мa becomes = HI= D.

....

(3.)

cos ZMα = sin D′ = sin D COS I

sinessin Esin zмa sin PM = cos D'cos (L).. (4.) where E is the height of the style, or elevation of the pole above the plane.

Then, pb being any hour circle whatever, the sphe rical triangle Pbм, gives us, from equa. (4) of spherical trigonometry [chap. vi. 23].

cot мb sin PMCOS PM COS Pмb + sin Pмb cot P. But, cos Pмb = sin d′, and sin Pмb = cos D ́';

COS

also, cot. Therefore, dividing by sin PM,

sin

COS D'

cot Mb = sin D' tan (L — 1) +

....

P (5.)

COS (L-1)

cot Ma,

tan MT = tan (Ma + 90°) = —

where MT is the inclination of the horizontal plane to the meridian.

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 land D'. Thus, for the first term,

sin / cos I cos L + sin 27 sin L cos 21 (tan / cos L + tan 2/ sin L)

The last values of sin D' tan (L — 1) and of

COS D'

COS (L

being substituted for them in equa. (5), it becomes

cot P (6.)

....

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 × 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

7),

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

tan Ms =

sin D cos I cot L+ sin Itan D

1 -tan I sec D Cot L

Farther, since cos D' cos (L

(7.)

-

=sin PMS sin PM

sin E

the equation (4) becomes

sin 1 (cos L cot l

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 10, or the plane become erect; then from equa. (6), we have!

(10.)

In this hypothesis equa. (7) becomes

tan MS Cot L sin D

cot s

By which the angle between the substyle and the hori

zon is determined.

And equa. (8) becomes

sin E COS 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 I become = O, 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 = 0, 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 = 0, we have I = 90°, the equations (6) and (8) become"

(19.)

or, tan мb = sin L tan P.... (18.)

and sin E = sin L

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