cation with the external air; the eudiometer is then rapidly depressed and closed. In this position the tube from the hydrogen can be rinsed again, independently of the eudiometer, so that the washing may be considered as complete and thorough.

The eudiometer being brought into connexion with the hydrogen is again raised, and 18 c.c. of hydrogen gas are taken in under atmospheric pressure.

The hydrogen kept over water is saturated, and a thermometer with its bulb in the bell-jar gives the temperature of the gas, which is very nearly that of the laboratory; so that by the time the gas is ready to be measured in the eudiometer it shows no tendency either to contract or dilate. The eudiometer now contains the volume of hydrogen required for the analysis, and the stop-cock is turned shutting off the gas from the holder, and opening the Vshaped tubes through and through in readiness for washing out with the air to be analysed.

The air from the large glass jar is introduced into the eudiometer in the following way. Having filled the funnel referred to above with water, the latter is let into the jar by opening slightly the pinch-cock closing the funnel; at the same time the glass jar having been connected with the V-shaped tube of the eudiometer by indiarubber tubing, is opened towards the instrument, when the air displaced by the water added rinses out the india-rubber and steel tubings. There is plenty of air in the jar, so that no necessity occurs to be saving; when the tubes are rinsed the eudiometer is raised in the mercury up to about 45 c.c., carrying a column of mercury with it; then the two-way stop-cock is very carefully turned so as to admit the air to be analysed, which is aspired by the mercury as it subsides. Thus some 27 c.c. of air are introduced. The aspiration must be fairly rapid, and the fall of mercury in the tube should be stopped by turning the stop-cock before the mercury has quite reached its level in the trough, otherwise there is a risk of a recoil of the mercury, and a "pumping" which it is important to avoid. The inixed gases are left undisturbed for two or three minutes, and their volume is read off under atmospheric pressure, the eudiometer being next moved up and down in the mercury by a few centimetres, so as to effect the perfect mixture of the gases. The instrument is now slightly raised, carrying with it a short column of mercury, and the gases are ignited by the electric spark under reduced atmospheric pressure. This mode of proceeding, recommended by McLeod,* weakens considerably the violence of the explosion, and ensures perfect safety. Immediately after the explosion the gas in the eudiometer is brought approximately under atmospheric pressure.

* McLeod, "On a new Form of Apparatus for Gas Analysis," 'Chem. Soc. Journ.,' 1869.

A slow contraction now takes place as the heat produced by the explosion is radiated from the instrument; it is advisable to wait about twenty minutes, until the contraction is complete, and the volume of the gas is read off under atmospheric pressure.

The instrument should be sheltered from any draught, or from the direct radiation of a fire, and indeed be kept from any change of temperature, and with that object I find it advisable to shelter it with a cardboard tubular shield sliding up and down the mercury trough.

I

If air taken directly from the atmosphere is to be analysed, in order to ensure its being saturated it will be advisable to pass it through a tube full of wet horse-hair, and obtain it directly from the tube into the eudiometer. In the above account of the manipulation required, the hydrogen is introduced before the air into the eudiometer. have tried to let in the air first, but this plan was not successful apparently because the mixture of air and hydrogen was incomplete before the explosion. The hydrogen being collected first in the eudiometer will rise from its comparative lightness as the air is drawn in and mix with it perfectly, while the stream is sufficiently rapid to prevent any of the mixture from diffusing out of the tube. It should be borne in mind that after a number of analyses the water resulting from the explosions accumulates on the surface of the mercury in the eudiometer, and the mercury meniscus is no longer clearly seen. This can be easily avoided by drying the tube with filtering paper after a certain number of analyses. The following are a few determinations of oxygen in atmospheric air made with the form of eudiometer described above. They are not selected, but given in succession in the order in which they were made. And I must here beg to record the valuable aid of my assistant, Mr. Charles F. Townsend, F.C.S., in the present inquiry.

One analysis omitted: obviously too high from insufficient rinsing.

XIII. "Theorems in Analytical Geometry." By W. H. L. RUSSELL, F.R.S. Received June 21, 1888.

To determine the envelope of the first polar of any curve, when th pole moves on a given curve of the third order.

dF

Let F (§, 1, 3) = 0 be equation to the surface; then if p = de'

dF

r = px + qy + rz = 0 is the equation to first polar,

when (x, y, z) moves on a given cubic,

x + y3 + 23 + 6mxyz = 0.

Then differentiating

(x2+2myz) dx + (y2 + 2mxz) dy + (z2 + 2mxy) dz = 0.

pdx + qdy + rdz = 0.

Then as usual

x2 + 2myz = λη,

y2 + 2mxz = xq,

z2 + 2mxy = λr.

Then eliminating z by the equation to the first polar, we have

....

Ax2+Bxy + Cy2 = p,

Da2 + Exy+Fy2 = 4,

Ga+Hay+ Ky2 = r;

where A, B, C . . . . are functions of pqr, whose forms are immediately seen, and the arbitrary multiplier is omitted because it will disappear in the final result: then we find at once the values of æ3, xy and so z2, y2, and therefore of x, y, z, which we may substitute in the equation to the polar, and so obtain the envelope. But we may find a more symmetrical result thus: eliminating as before by means of the equation to the polar

A'y2+ B'yz + C'z2 = p,

D'y2 + E'yz + F'z2 = 9,

G'y2 + H'yz + K'z2 = r;

and moreover