the most polluted waters, is stated to be the mixing in the
small tanks the necessary chemical reagents, at the commencement of
the working day; and at the close of the day the opening of the mud
cocks shown in our engraving, to remove the collected deposit upon the
plates. For the past six months this system has been in operation at a
dye works in Manchester, successfully purifying and softening the
foul waters of the river Medlock. It is stated that 84,000 gallons per
day can be easily purified by an apparatus 7 feet in diameter. The
chemicals used are chiefly lime, soda, and alumina, and the cost of
treatment is stated to vary from a farthing to twopence per 1,000
gallons, according to the degree of impurity of the water or sewage
treated.

The results of working at Manchester show that all the visible filth
is removed from the Medlock's inky waters, besides which the hardness
of the water is reduced to about 6 deg. from a normal condition of about
30 deg.. The effluent is fit for all the varied uses of a dye works, and
is stated to be perfectly capable of sustaining fish life. With
results such as these the system should have a promising future before
it in respect of sewage treatment, as well as the purification and
softening of water generally for industrial and manufacturing
purposes.--_Iron._

[Illustration: WATER SOFTENING AND PURIFYING APPARATUS.]

* * * * *

THE TRISECTION OF ANY ANGLE.

By FREDERIC R. HONEY, Ph.B., Yale University.

The following analysis shows that with the aid of an hyperbola any
arc, and therefore any angle, may be trisected.

If the reader should not care to follow the analytical work, the
construction is described in the last paragraph--referring to Fig. II.

Let a b c d (Fig. I.) be the arc subtending a given angle. Draw the
chord a d and bisect it at o. Through o draw e f perpendicular
to a d.

We wish to find the locus of a point c whose distance from a given
straight line e f is one-half the distance from a given point d.

In order to write the equation of this curve, refer it to the
co-ordinate axes a d (axis of X) and e f (axis of Y), intersecting
at the origin o.

Let g c = x

Therefore, from the definition c d = 2x

Let o d = D
[Hence] h d = D-x

Let c h = y
[Hence] (2x) squared = y squared + (D-x)