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And the equation of the cone is


Or substituting and expanding,

(ad2 —z3)b3 — (a2d— B3)c3 — d(BS—y2)ab2+a(ay—B2)dc2 +d(BS—y2)a2c—a (ay — B2)d2b+28(ay—ẞ2)ac2—2a(BS—y2)db2

—y(BS—y2)a2d+B(ay—B2)ad2+3y(By—ad)b2c—3ß(By—ad)bc2 +d(By―ad)abc-a(ßy—ad)dbc+(ayd−3ßy2+2ß28) abd

- (aßd—3ß2y+2ay2)acd=0.

Now putting

bd-c2=p, bc-ad=q, ac-b2=r,

and in like manner

Bd—y2=P, By—ad=Q, ay—B2 = R,

and reducing, the final result may be expressed in the form

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Qabp+(ya—2ßb)q+ (Sa—yc)r}

acp+(yb Be)g + Sbr}




where a, b, c, d are current co-ordinates, and p, q, r are quadratic functions of a, b, c, d. The equation is (as it should be) satisfied by the equations (p=0, q=0, r=0) of the given curve; it is also satisfied per se when P=0, Q=0, R=0, i. e. when the vertex is a point on the curve; this indicates a change in form of the equation, and in fact the cone is in this case of the second order only. Suppose that the co-ordinates of the vertex are in B_Y 1 this case given by


B γ

==== (σ an arbitrary quantity), it



may be easily shown that the equation of the cone is

or at full length,


(bd—c2)+o(bc—ad) + o2 (ac—b2)=0.

In fact this equation is evidently that of a surface of the second order passing through the curve; and there is no difficulty in showing that it is a cone.

2 Stone Buildings, May 1, 1856.

III. On a peculiar Power possessed by Porous Media (Sand and
Charcoal) of removing Matter from Solution in Water. By
HENRY M. WITT, F.C.S., Assistant Chemist to the Govern-
ment School of Applied Science*.


ARIOUS methods have been employed at different times for the purification of water for the supply of towns, but none has been found so practically convenient and efficacious as simple filtration through porous media, such as sand; moreover, charcoal being known to possess a peculiar power of removing organic matters from solution, this substance has been suggested and occasionally employed either as a substitute for, or an auxiliary to, ordinary sand filtration.

The following experiments were undertaken with the view of ascertaining by chemical analysis the more precise nature of the effects produced upon ordinary river-water, such as that of the Thames, by its passage through filters composed of these media respectively, and of comparing their powers; but it is believed that the results obtained possess an interest extending considerably beyond the question to assist in the solution of which they were made.

Before proceeding to the construction of experimental filters, I availed myself of the kindness of Mr. James Simpson, engineer to the Chelsea Waterworks Company, to investigate the results obtained by that Company's system of filtration as carried on up to the present time at Chelsea; and I have much pleasure in embracing this opportunity of expressing my obligation to this gentleman, and specially also to his son, Mr. James Simpson, jun., to whom I am greatly indebted for his very able cooperation throughout this inquiry.

The system of purification adopted by the Chelsea Waterworks Company at their works at Chelsea, consisted hitherto (for the supply has by this time commenced from Kingston) in pumping the water up out of the river into subsiding reservoirs, where it remained for six hours; it was then allowed to run on to chalin the filter beds. These are large beds of sand and gravel, each exposing a filtering surface of about 270 square feet, and the water passes through them at the rate of about 64 gallons per square foot of filtering surface per hour, making a total quantity 104/87 of 1687-5 gallons per hour through each filter.

The filters are composed of the following strata in a descending order :

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Ground line.

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4. Fine gravel

5. Coarse gravel

These several layers of filtering materials are not placed perfectly flat, but are disposed in waves, as seen in the sectional drawing; and below the convex curve of each undulation is placed a porous earthenware pipe, which conducts the filtered

water into the mains for distribution.

Chelsea Waterworks, Thames Bank. Transverse Section of Filter.
Downward Filtration.


a. Water line.

b. Top of fine sand.

c. Collecting drain (perforated).

Note.-Depth of water over the sand =4 ft. 6 in.

The upper layer of sand is renewed about every six months, but the body of the filter has been in use for about twenty years.

Samples of water were taken and submitted to examination— 1st. From the reservoir into which the water was at the time being pumped from the middle of the river.

2nd. From the cistern after subsidence and filtration.

Experiments of this kind were made on three different occasions, viz. on the 12th of September and the 29th of December, 1855; and also on the 10th of May 1856; and the results are embodied in Tables I., II. and III., each containing four columns,-No. 1 showing the quantities of the several substances originally present, represented in grains in the imperial gallon (70,000 grains) of water; No. 2, the amounts present after filtration; No. 3, the actual quantities separated in grains in the gallon of water; and No. 4, the per-centage ratio which the amounts separated bear to the quantities originally present.

Results of Filtration through Sand at the Chelsea Waterworks. TABLE I.-September 12, 1855.

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By these analytical results (if entitled to confidence, and I may state that all due care and attention was given to ensure their accuracy) it is shown that this process of mere sand filtration is one of more importance, and of a more peculiar and interesting character, than has generally been supposed.

It has been asserted as a principle that sand filtration can

only remove bodies mechanically suspended in water, but I am not aware that this statement has been established by experiment; in fact, I am not acquainted with any published analytical examination of the effects of sand filtration.

These experiments supply the deficiency, and show moreover, that these porous media are not only capable of removing suspended matter (80 to 92 per cent.), but even of separating a certain appreciable quantity of the SALTS from SOLUTION in water! viz. from 5 to 15 per cent. of the amount originally present, 9 to 19 per cent. of the common salt, 3 per cent. of the lime, and 5 of the sulphuric acid.

It is curious also that the proportion of matter removed in this way depends to a certain extent upon the degree of impurity of the water; the greater the quantity of matter originally present in the water, the larger the per-centage ratio of the salts removed, e. g. :

In Sept.


In Dec. 31.467

In May.



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Total impurity of water. Per-centage of salts removed This point will, however, become more apparent by comparing these experiments with those made at Kingston, where the water is much purer, to which I shall have occasion hereafter more particularly to refer.

This fact, of the power possessed by sand and other porous media, which rests for demonstration not only upon the three preceding experiments, but also upon others to be presently described, is one of great importance, not only in a hygienic and economic point of view as relating to the great question of water supply, but also in its bearings upon agriculture and geology.

It is possible that soils may remove matters from solution in water, not only by decompositions between the contained salts and the aluminous silicates of the soil (as demonstrated by Mr. Way), but also in virtue of this peculiar action.

Again, water containing considerable quantities of saline matter in solution may, by merely percolating through great masses of porous strata during long periods, be gradually deprived of its salts to such an extent as probably to render even sea-water fresh.

This may in fact be one of the causes contributing towards the production of freshwater springs which ebb and flow with the tide in the vicinity of the sea: for instance, Darwin, in the 'Voyage of the Beagle' (vol. iii. p. 545), mentions that on Keeling Island, one of the coral reefs near the coast of Sumatra, there are freshwater wells which ebb and flow with the tide. Mr. Darwin, however, suggests another explanation of this

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