Professor E. E. Barnard assisted by Mr. Ritchey made 7 exposures with a 6-inch lens of 62 feet focus mounted horizontally. The light was thrown into the telescope by a coelostat with a 12-inch mirror made by Mr. Ritchey. The exposures made varied from one-half second to thirty seconds in duration, three upon 14 x 17 plates and four upon 25 x 30 plates, the largest that have ever been used for photographing the corona, with exception of those used at the same station by the Smithsonian Institution. A newspaper report, June 14, states that one of these plates, the last exposed, duration 1 second, had been developed and brought out many interesting details close to the edge of the moon. Mr. Barnard's hope is to bring out upon the longer exposed negatives the details of the outer corona as well as the inner upon that magnificent scale, which makes the diameter of moon's disk 7 inches; and we may be sure that Mr. Barnard will do it if any man can.

Several smaller cameras, from a 6-inch Voightlander down, were operated by volunteers. Some were mounted equatorially and driven by clockwork, others were simply fixed upon posts.

Professor E. B. Frost assisted by Dr. Isham of Chicago photographed the spectrum, using three slitless spectroscopes, one with a train of three prisms, one with a single prism and one with a concave grating. The signal for photographing the "flash" spectrum was given by Professor Frost observing with a flat grating. He said that he saw only dozens of lines reverse where he expected to see hundreds. Elsewhere also the "flash" appears to have been disappointing. At Pinehurst the whole party depended upon the observation of this phenomenon for the signal for the beginning of their program during totality. The one deputed to make this observation and give the signal failed to see the "flash" and as a result several seconds of totality had elapsed before the signal was given.

The coronium line too appears to have been weaker than usual, and Professor Young who made the determination of the position of this one line his special work failed to see it.

Professor Hale assisted by Mr. Ferdinand Ellerman undertook the very delicate operation of measuring the heat of the corona with bolometric apparatus, in connection with a large siderostat kindly loaned by the Smithsonian Institution. He had a very unfortunate experience. On arriving at Wadesboro ten days before the eclipse he found that a very delicate part of the apparatus had been broken. He had been warned by previous experience and had shipped a lathe and all necessary tools as a part of the

equipment of the expedition. With characteristic energy he set about constructing a new bolometer and succeeding in making a better one than he had before and performed satisfactory preliminary experiments. All was in perfect adjustment and ready for use during totality, when just as the important moment arrived a small stick, which was used for some purpose in the dark bolometer room and had been leaned against the wall, fell and threw the galvanometer needle out of balance. The operation of balancing requires usually from two to three minutes. Professor Hale said that he never worked harder in his life than during the next minute and a half, and he succeeded in getting the instrument into balance, but only to see the sunlight reappear before any measures could be taken.

Mr. C. G. Abbott of the Smithsonian Institution by similar methods succeeded in detecting a slight amount of radiation from the corona as compared with that from the black disk of the moon. Professor Hale is confident from the results of his preliminary experiments that by these methods he can detect the change of heat at the edges of the great coronal streamers in full sunlight.

Professor A. S. Flint of the Washburn Observatory made the observations for time and position at Wadesboro, and noted the times of contact and counted the seconds of totality for Professor Frost.

The longest focus telescope ever used in observing an eclipse is probably that used by the Smithsonian Institution at Wadesboro, which had a focal length of 135 feet. This would give an image of the moon nearly 15 inches in diameter.

The telescope was mounted horizontally, the light being furnished by a coelostat. In the Chicago Journal of June 15, we find this statement concerning the photographs taken with this instrument:

"Mr. Smillie exposed six 30 x 30 plates during totality, with times ranging from one-half second to sixteen seconds. All these negatives have not yet been developed. Those of one-half second, two seconds and four seconds exposure have been hurriedly examined, however, and they give clear indications of the crossing and recrossing of filaments like the appearance of a field of grain bending in the wind. The prominences and polar streamers appear in imposing magnitude and detail."

General Perturbations and Perturbative the Function. 309

GENERAL PERTURBATIONS AND THE PERTURBATIVE

FOR POPULAR ASTRONOMY.

FUNCTION.

J. MORRISON, M. A., M. D., PH. D.

It was first proved by Sir Isaac Newton in his immortal work The Principia, that when a particle moves around a centre of force which varies directly as the mass and inversely as the square of the distance, the path or orbit described is a conic section with the centre of force in the focus, and the radius vector describes equal areas in equal times. In the case of a planet moving. around the Sun, the orbit is an ellipse with the Sun in one of the foci. It is evident that the planet will move most rapidly in perihelion and most slowly in aphelion. The mean motion or that which it would have, if it described a circular orbit, is evidently 2π equal to 360° or 27 divided by the periodic time Tor From Τ

perihelion to aphelion the true place of the planet will be in advance of the mean place; at aphelion the mean and true places will coincide, and from aphelion to perihelion the mean place will be in advance of the true until perihelion is reached when they again coincide and so on.

The angular distance between the true and mean places or to express it more technically, between the true and mean anomalies, is called the elliptic inequality or 'the equation of the centre,' and it is the only correction to be applied to the mean to obtain the true anomaly in the case we are now considering.

If however we suppose another planet to be added to the system, the circumstances of the motion of both planets, become much more complicated; each disturbs the motion of the other; the equable description of areas which obtained in the case of a single planet now no longer exists, and the computation of the true place of either planet is a work of prodigious difficulty. It is the famous "problem of three bodies" which has severely taxed the ingenuity and analytical skill of mathematicians since the discovery of the law of universal gravitation.

In this and subsequent papers we purpose to develop as clearly and as briefly as the difficulties of the problem will admit, the formulae for undisturbed and disturbed motion, so as to enable the reader to understand the more abstruse and elaborate developments of La Place, LeVerrier and others.

Let x, y, z be the coördinates of a planet referred to the centre

of gravity of the Sun S, as the origin and r its radius vector, also let m denote the ratio of the mass of the planet (m) to mass of planet mass of planet that of the Sun or m = mass of Sun

=

k2

where k2 is the well known Gaussian constant of solar attraction whose value will be determined farther on, then the mass of the planet = mk2.

Let x', y', z' be the coördinates, r' the radius vector and m'k2 the mass of a second planet (m') and similarly for other bodies of the system and let P, P1, P, be their mutual distance, or mm', mm" etc., then we shall have

Sm2 = x2 + y22 + z'

12

r" Sm" = x222 + y'"2 + z'2, etc.

and p2 mm"2 = (x' - x)2 + (y' − y)2 + (z′ — z)2

−

p1 = mm'"2 = (x" - x)2 + (y" — y)2 + (z"-z)2, etc.

(1)

Considering only three bodies, the Sun S, the disturbed planet (m) and the disturbing planet (m') and putting for the sake of brevity k2 + k2m or k2 (1 + m) = μ, it is evident that in the relative motion of (m) around S, it will be acted on by the three μ k2m' k2m' forces r'' p2 r/2

and

respectively directed along the lines

mS, mm' and m'S, and since the cosines of the angles which the directions of each of these forces make with the axis of x, are respectively

We shall have for the components of these forces parellel to the same axis

the first and third being negative because the force tends to di

minish the coördinates x, y and z.

For the components of these same forces parallel to the axis of Y we shall have in a similar manner

The sum of these three components for each direction is evi

dently the total force parallel to this direction, which acts on the planet (m) and it must be equal to the accelerations

d2x d'y

dt" dt2 d2z and respectively. If then we extend these results to a numdt2

ber of bodies m", m'", etc., we shall have for the equations of the relative motion of (m) around S.

where the symbol indicates that each mass k2m”, k2m’”, in

troduces a term similar to that which results from the action of m' on m - a term which we obtain by simply changing in the second members m', p, x' and r' into m", p', x'' and r" respectively, and so on.

To facilitate the solution of the problem, the second members of (2) are put into the following form.

gard to the variables x, y and z. If then we put

we shall have for the partial differential coefficients with respect