obliquity of the ecliptic at any time, the rate of the luni-solar precession at that time during a Julian year, will be represented by

c cos w

where c 54."94625 nearly.

Now let ON'N be the fixed plane of reference which may be

either the ecliptic at a given epoch or, still better, the invariable. plane of the system, or any other arbitrary fixed plane.

N'E the position of the ecliptic

and NE that of the equator

at any time t

so that the point E is the autumnal equinox at that time. ON=4, ON'=', O being a fixed point, and 'the incli nation of the equator and ecliptic respectively to the fixed plane, and the angle N'EN, or the obliquity of the ecliptic at time t. Also let NE=λ.

Then the quantities

ptan 0' sin y' and q tan e' cos y'

are known in terms of t from the theory of the secular variations of the plane of the earth's orbit, and e' may be considered as a small quantity of the 1st order, the square of which we propose to take into account.

In the triangle N'EN we have

sin

cos @ = cos / cos 0' + sin 0 sin 0' cos (y — 9')

cos sin cos 0'

sin o sin

which give

=

= sin 0' sin (ø — y')

and λ when 0 and are known.

From the instantaneous motion of the equator with reference to the ecliptic at time t, supposed for an instant to be fixed, it is

easily seen that we have

аф

d t

sin e

or, substituting from above for cos w, sin w cos and sin w sin ▲

de

d t

=c cos2 0' [cos 0 + sin 0 tan 0' cos (y-')] tan 0'sin (y—y') which are the differential equations for determining 0 and 9, 0' and

being supposed to be already known in terms of t.

From the above we may deduce the following:

dp

= (sin 0 cos )+(sin 0 sin ).

d

dq (COS)

dt

dt

The integration of these equations may be readily effected by the method of indeterminate coëfficients.

Suppose the values of p and q to be

p = 27, sin (g, t + ß;)

ΣΥ

q=r, cos (g, t + B1)

where i takes the successive integral values 0, 1, 2, etc., equal in number to the number of planets considered, and the quantities

and

Vi gi B are known constants.

་

Then we may find that

0 h + tan ha, (a, -1) 72+ coth 2 (a) r?

=

7; cos
cos (9i
{

4 = kt + a + b, 7, sin (kg) t + a — B; }

(7) sin2 { (k − g;) t + a — B1}

sin

{(2 kg-g;) t + (2 a — ß; — B;)} sin (99) t+B ; — B; }

{

i

in which i and j are supposed to be different integers.

b=a2 (a,1) tan3h+a, (a,2 + a, − 1) + ža2 cot 2

ii

+a; (a; -1]) tanë h

k

[a; +a;] cot2 h

k a1.. tan h +

+a; (a; −1)] tan2 h

or the value of this last coefficient may be otherwise expressed thus

+[(a,-1)+(a,− 1)] [(a,− 1)+(a,— 1) — 1]; Also the value of the obliquity of the ecliptic is thus expressed in terms of the same quantities

w = h + 2 (a; —1) 7, cos {(k-g;) t + a — ß; }

Ti

r2

¿ [-a+, (a,1)2 tan h—(a,-1)2 cot h] r2 cos 2

—Σ { } π; — 9; [a2; — a2; — 2a, + 2a,] tan h

[aa] cot h― [† a; (a; — 1)

+a; (a,1)] tan h — [± a; +a; — ] cot h

i

+rir; cos {(9-9;) t + ß; — B; }

Also the value of k in terms of the constant c which, as stated before, is known from the theory of precession is

k = c cos h {1- Σ † (a; — 1) (3a, —5) r?}

h and a are the arbitrary constants which enter into the complete integrals of our equations, and they are to be determined so as to make the initial values of 0 and y, or those of w and equal to the observed values.

It is to be remarked that one of the values of g is 0 and if the invariable plane of the system be taken as the fixed plane of ref erence, the corresponding value of q will be also zero, so that the expression for 0, and will be considerably simplified by this

choice of the fixed plane.

According to Stockwell's determination, in Vol. 18 of the Smithsonian Contributions, the longitude of the ascending node of the invariable plane on the ecliptic of 1850 is 106° 16' 18", and the inclination to the same ecliptic is 1° 35′ 18.9"

= =

Also, as already mentioned, if we make the invariable plane of the system our plane of reference, we have for g, 0, 7.0, and the remaining values of g, and those of log 7, which correspond to them, according to Stockwell's determination, will be the following:

i=1

i=3

i-4

i=5

i=2 Ji-2.''9161 -25.''9350 -5.''21365 -6.''6693 -17.''6266

i=6 i=7 -18.''9365 -0.''66166

where the quantities g, are expressed in seconds and have reference to a Julian year as the unit of time, and the quantities 7, are expressed in the circular measure.

Hence we may find, with reference to the invariable plane, for the epoch 1850

023° 3' 42."7

- 4' = 257° 20' 21."5

Now, in the figure before given, the point N' is the descending node of the invariable plane on the ecliptic of 1850, so that the longitude of N' is 286° 14' 118"

Also the longitude of the point E which is the autumnal equinox, is 180°. Hence N'E= 253° 45' 54"

4'257° 20′ 21."5

&= = 183° 34′ 39."5

Also, according to Stockwell, the obliquity of the ecliptic in 1850 was

whence by substitution all the terms in 0, and may be found numerically.

NOTE ON NEWTON'S THEORY OF ATMOSPHERIC REFRACTION, AND ON HIS METHOD OF FINDING THE MOTION OF THE MOON'S APOGEE. By Prof. J. C. ADAMS, Cambridge Observatory, England.

[ABSTRACT]

Professor Adams exhibited photographs of some of the MSS. of Newton preserved in the collection lately presented to the University of Cambridge by the Earl of Portsmouth.

These papers show that Newton had thoroughly mastered the theory of atmospheric refraction and that he was acquainted with the true principles on which the motion of the moon's apogee should be found and that he had obtained a fair first approximation to the amount of that motion.

THE NEBULE. BY LEWIS SWIFT, Director of the Warner Observatory, Rochester, N. Y.

FOR many years it has been the opinion of astronomers that the search for new nebulæ, since it was abandoned by the Herschels, has been greatly neglected. Entertaining this view myself also, I, as soon as the 16 in. refractor of the Warner Observatory was mounted, put the idea into practice, and since then, excepting the time devoted to comet-seeking, have made their discovery and observation my special work. The success thus far attending the effort awakens the suspicion that the number of undiscovered nebulæ is very great and perhaps equalling-it may be surpassing- those already known. Only telescopes of the largest aperture, equipped with eye-pieces especially adapted for the investigation, can work successfully in this interesting field, hence these bodies fail to arouse the popular enthusiasm that the double-stars, the sun, the moon etc., inspire, all of which can be observed with smaller instruments. In 1784, just one hundred years ago, the entire number of nebulæ known was less than 150. Their numbers, at various epochs, were as follows; in 1612 only the great nebula in Andromeda — the most conspicuous naked-eye nebula visible from our latitudes

had been observed. Forty-four years later, in 1656, another had been added, viz.: the great nebula in Orion, discovered by