ragged) fall of the atmospheric pressure, which reached its minimum about 4h 45m P.M. There was then a very abrupt and nearly perpendicular rise of about five hundredths of an inch of pressure, or rather less, after which the rise still went on, but only more gradually.

Through the kindness of the Rev. R. Main, of the Radcliffe Observatory, I have been favoured with a copy of the trace afforded by the Oxford barograph during this squall, in which there appears a very sudden rise of nearly the same extent as that at Kew, but which took place about four o'clock, and therefore, as on the previous occasion, somewhat sooner than at Kew. This change of pressure at Oxford was accompanied by a very rapid fall of temperature of about 8° Fahr.

The minimum atmospheric pressure at Kew was 29.52 inches, while at Oxford it was 29.28 inches.

It will be seen from the Plate that at Kew the electricity of the air fell rapidly from positive to negative about 4h 30m P.M., and afterwards fluctuated a good deal, remaining, however, generally negative until 5h 22m P.M., when it rose rapidly to positive.

We see also from the Plate that there was an increase in the average velocity of the wind at Kew during the continuance of this squall. To conclude, it would appear that in these two squalls there was in both cases an exceedingly rapid rise of the barometer from its minimum both at Oxford and at Kew, this taking place somewhat sooner at the former place than at the latter; and that in both cases the air at Kew remained negatively electrified during the continuance of the squall, while the average velocity of the wind was also somewhat increased.

The Society then adjourned over the Christmas recess to Thursday January 7, 1864.

"On the Equations of Rotation of a Solid Body about a Fixed Point." By WILLIAM SPOTTISWOODE, M.A., F.R.S., &c. Received March 21, 1863.*

In treating the equations of rotation of a solid body about a fixed point, it is usual to employ the principal axes of the body as the moving system of coordinates. Cases, however, occur in which it is advisable to employ other systems; and the object of the present paper is to develope the fundamental formulæ of transformation and integration for any system. Adopting the usual notation in all respects, excepting a change of sign in the quantities F, G, H, which will facilitate transformations hereafter to be made, let

A=Inty'+'),

-F=Ymy,

B=2m(+),
-G=Ymx,

C=2m(x2+y3),
-H=Σmry;

* Read April 16, 1863: see abstract, vol. xii. p. 523.

and if p, q, r represent the components of the angular velocity resolved about the axes fixed in the body, then, as is well known, the equations of motion take the form

To obtain the two general integrals of this system: multiplying the equations (1) by p, q, r, respectively adding and integrating, we have for the first integral

Ap2+Bq2+Cr2+2(Fqr+Grp+Hpq)=h,

(2)

where h is an arbitrary constant. Again, multiplying (1) by

Ap+Hq+Gr,

Hp + Bq +Fr,

Gp +Fq+Cr,

respectively adding and integrating, we have for the second integral

(Ap+Hq+Gr)2+(Hp+Bq+Fr)2+(Gp+Fq+Cr)2=k2,

where k2 is another arbitrary constant. This equation may, however, be transformed into a more convenient form as follows: writing, as usual,

A=BC-F2, 3=CA-G', C=AB-H2,
J-GH-AF, G=HF-BG, H-FG-CH,
A+B+C=S,

and bearing in mind the inverse system, viz

▼= A H G

HB F. (4)
G F C

VA=BC-F2, VB=CA-G2, VC=AB-H',

VF=GH-AF, VG=HF-BG, VH=FG-CH,
A+B+C=S,

we may transform (3) into the following form :

(

(AS-B-C)p2+2(FS+)qr

+(BS-C-A)q'+2(GS+G)rp
+(CS−A−B)r2+2(HS+H)pq=k2,

which in virtue of (2) becomes

(6)

(A —§)p2 + (B—§)q2+(C−S)r2+2(Fqr+Grp+Hpq)=k2—Sh. (7) This form of the integral is very closely allied with the inverse or reciprocal form of the first integral (2), and is the one used below.

In order to find the third integral, we must find two of the variables in terms of the third by means of (2) and (7), and substitute in the corre