EXAMPLE III. Let a 1820, and b=1660750;

b

b

in

then=912, and a+ -—-—=27321⁄2, and a-907; and 2732 divided by 907 gives 3 the quotient, and 10 in the remainder, thus 907) 2732(3

fo that 3 is the common ratio, and 5 the first and the feries is 5, 15, 45, 135, 405,

term;

1215.

EXAMPLE IV. Let a=75, b=2125; then

b

=3

=28 and a+===103; divided by a-—-— 463, gives 2 in the quotient, and 10 in the remainder; fo that the series is 5, 10, 20, 40.

It remains that something be said of the series x-xr+xr2 —xr3, &c. ad infinit. Here is r negative; and fince

x

=

the fum of the series x+xr

by

+xr2+xr3, &c. ad infinit. consequently, changing the fign of the quantity r, i. e. by

ries x—xr+xr2 —xr3, &c. ad inf. will be

I + r

If in this feries r-1, then the feries is x-x+x

-x+x, &c. ad infinitum, and its fum=

x

2.

I have added this laft feries, x-x+x-x, &c. because it has lately been the fubject of much debate, and I have determined its fum by a different method to others. Mr. Vince is right in calling the fum of this feries, though the method by which he made his deduction was false, and which gave his opponents the opportunity of faying, that the fum of the feries might have

On HALOS, by the Rev. JAMES WOOD, A. M. Fellow of St. John's College, Cambridge. Communicated by Thomas Percival, M. D. F.R.S. &c.

READ OCTOBER 12, 1787.

DIFFERENT hypothefes have been made

ufe of by Des Cartes, Huygens and Newton, to account for the appearance of Halos, or coloured circles, round the fun and moon. The first of these supposes rays of light to be refracted by pieces of ice formed like double convex lenfes, which however, he confeffes, we never find upon the furface of the earth. Huygens fays that halos are caufed by fmall globules of fnow, furrounded each by a fhell of water; the

rays

rays which fall upon the fnow being stopped, whilst those which fall on the water are refracted into their respective colours. Newton accounts for the appearance, by fuppofing that rays of light, in paffing through globules of water, are fometimes in a difpofition to be reflected, and fometimes to be refracted. All these fuppofitions how. ever seem to be unfatisfactory, as the rays will not fall upon the eye in fufficiently strong pencils.

The order of the colours in these crowns, as laid down by Des Cartes and Huygens, is contrary to that which Sir Ifaac Newton gives.

If they judged from fuch as are commonly feen round the moon, they might eafily be deceived, as the colours are in general not very diftinct: the halos I have obferved have uniformly been fimilar to those described by Newton, at the end of his fecond book of optics.

Cambridge, Nov. 30, 1786. Three very brilliant halos appeared round the moon: the order of the colours, beginning from the moon, was, white of confiderable breadth, yellow, orange, red; violet, green, yellowish, red; violet, green, yellowish, red. The red of a fourth was fo faint as to be scarcely vifible. The radii of these rings might be about one, two, and three degrees, but they were not measured. The red and violet in each cafe feemed contiguous.

Dec. 2d. A fingle halo appeared round the moon its djameter was about 2, and the colours

Ꮓ