A3= — §¥3d0= — 1(sin 0'

&c.

A‚= — Sr,do = −1(sin 0day+n.n+1 sin 0. A,. de

ᏧᎾ +n.n+1 sin 0. Aq. do

or, substituting for A and performing the integrations,

dA,

de), (2)

dA,

Ssin •),

&c.;

+(c.n.n+1(n.n+1−1.2)+c2.n.n+1) cos✪+Cg

If the condition be imposed that the series shall terminate, it may be satisfied by means of the constants introduced in the subsidiary integrations.

* Instead of taking for the base of the arbitrary function the above value of u, we may take eu for the base where u has the above value, the form so obtained being identical with that given by Professor Donkin. Its adoption, however, adds materially to the complexity of the result.

To effect this, let Am be the coefficient of the last term of the series for w; then from what has preceded it is evident that we may assume

m

Am =am cos +am-1 cosm1+am-2 cos 0-2+ &c.,

where am, am-1, &c. are constants; and since by hypothesis. Am-1=0, the formulæ (2) give us

dAm de

0= sin 0 +n.n+1 Ssin 0. Am do.

The substitution in this equation of the above expression for Am gives

0=mam+1 cos 0m+1 +m−1am-1 cos 0" +m−2am−2 cos

m-2

+m-3am-3 cos 0m2 + &c.

m-1

·m—1am-1 cos 0m2 + &c.

(n.n+1-n-1n-2) (n. n+1-n-3n-4)

&c.,

am-4=

&c.

ат,

from which the law of formation of the terms in the series for Am or An is obvious. These being known, An-1 can be determined by means of the equation

or

A„= −1(sin @dan-1 +n. n+1ƒsin 0 A-do

de

A2-de),

0=2A,+ sin dan1+n. n+1ƒ sin 0 A-do; . .
0.

dAn-1 de

for, from what has preceded, it is evident that we may assume

-1

n-3

-5

(3)

An-1=bn−1 cos 0|^▬1 +·bn−3 CoS On−3 +bn−5—cos0|”−5 + &c. (where b-1, bn-3, &c. are constants); and substituting this value in (3), we shall get

0=2a, cos 0"+2an-2 cos 0"

n-2

n-4

+2an-4 cos 04+ &c.

+n−1 bn-1 cos 0"+n−3bn-3 cos 02-2+n−5 bn-5 cos 0]”−*+&c. -n-1bn-1 cos - -n ·n-3bn-3 cos 0"

in-2

-&c.

whence, equating to zero the coefficients of the different powers of cos e, we derive the following:

in-2

n-4

An-2=Cn-2 COS 0\"2+Cn-4 COS +Cn-6 cos 0-6+ &c.,

we may determine the coefficients Cn-2, Cn-4, &c. by means of

the equation

or

An-1=

de

+n.n+1 S sin 0 An-2d0
Ssin e An-de),

de

+n.n+1 S sin An-2d0 ;

0=2An-1+ sin @dAn-2

Hence, ing T for log, (tan), and (u) for "dun. X(u),

we have the following expression for w, viz.

where, if we choose to put an = 1, as we may do, we shall have

an = 1,

An-2=

an-4=

n.n 1

n.n+1—n−1n-2′

n.n-ln-2 n-3

(n.n+1-n-1n-2) (n.n+1-n-3n-4)'

an-6=

n.n-1n-2n-3n-4n-5

(n.n+1−n—ln−2)(n.n+1−n−3n—4)(n.n+1−n—5n—6

and where the constants b-1, bn-3, &c., Cn-2, Cn-4, &c., taken in order, are determined by equations (4) and (5), &c. in terms of known quantities.

If n be even, the last two terms of the series will be of the form

2-1)

(n-1

+(h cos20+ k) {("-") (T+ $√ −1)+"' (T−√ −1)}

(n)

+7{Y("T+$√ −1)+↓% (T−$ √ −1)}.

If n be odd, the last two terms will be

+41{y") (T+$√=1) +¥?") (T

where h, k, l; h, k, l, are known constants.

Lincoln's Inn, November 12, 1870.

LIV. On the Meteor of November 19, 1870. By W. J. MACQUORN Rankine, C.E., LL.D., F.R.SS.L. & E. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN,

ACCOUNTS have appeared in the Scottish newspapers of

a very large and bright meteor seen on the 19th instant, about 9 P.M. Greenwich time, from Edinburgh and from Carnwath (about thirty miles to the south-east of Glasgow). The meteor, as seen from Carnwath, is described as having passed from north-east to south-west, nearly overhead, and as having been followed by a rumbling sound after an interval of ninety seconds.

In the immediate neighbourhood of Glasgow there was, on that night, a haze so thick as to conceal the stars; but the glare of light produced by the meteor was distinctly seen at 30 seconds before 9, Greenwich time. It lasted three or four seconds, and, judging by the distinctness with which it illuminated terrestrial objects, was considerably brighter than the light of the full moon. From the appearance of the sky in the quarter in which the light vanished, the luminous object seemed to disappear in a southerly direction, at an altitude less than 30°.

A rumbling sound followed, after an interval which was not accurately ascertained, but is believed to have been between three and five minutes, corresponding to a distance of between thirty-six and sixty miles.

I am, Gentlemen,

Your most obedient Servant,

Glasgow, November 23, 1870.

W. J. MACQUORN RANKINE.