rough, and sulphureous taste, which reddened iris paper, which grew turbid, and deposited sulphur on standing, and, when boiled, gave off sulphurous acid, and finally which, saturated with ammonia and concentrated by evaporation, acted upon every re-agent exactly as a solution of hyposulphite of ammonia would have done. But the acid so obtained was in such small quantity compared with that contained in the metallic salt, that I will not venture to speak decidedly on the subject, but propose the earliest opportunity to try, as a last resource, the action of the voltaic pile.

SLOUGH, Jan. 8. 1819.

ART. III.-Application of the Indeterminate Analysis to the elimination of the unknown quantities from two Equations. By WILLIAM WALLACE, ESQ. F. R. S. Edin. Communi

cated by the Author.

WHEN two unknown quantities are to be determined from

two equations, one of which is of the second order, we may, by known theorems in the indeterminate analysis, eliminate both the quantities, and derive from them a new equation containing only one unknown quantity, of which each of the other unknown quantities is a given function.

1. For example, let us suppose that the equations are,

ax2+bxy+cy2 = no.... .... (2)

This last is taken of the second order for the sake of brevity, but it might be of any order whatever.

Thus, it appears, that the first equation is satisfied by the nature of the functions x and y, independently of any particular value of r and s.

of x and y being now substituted in the second equations, it be

an equation of the fourth degree, by which v may be determined, and thence the values of x and y.

2. Let the equation, which is of the second degree, be

2

x2+axy+by=c2.

..(1)

and let the other be

x2 yẞ+A xv y3 +&c. O...............(2)

an equation of any order whatever.

In this case, we must make

x=

(2 rs+as2) c
y=
r2+ars+bs=" r2+ars+bs??

and then, independently of any particular values of r and s, we shall have x2+axy+by 2=c2; if we now make s=1, (which comes to the same thing as to substitute =) we shall have x and y expressed by functions of r only; and these being substituted for x and y in the second equation, the result will be an equation involving only v: this being resolved, will give v, and thence x and y.

3. The most general equation of the second degree may be put under this form:

x2+axy+by+cx+dy=e.

This, by the usual transformations taught in books on Algebra, may be changed to

where x and y denote certain functions of x and y, which are at least rational in respect of these quantities, although they may contain known irrational numbers. This equation will be satisfied, if we make

The values of x and y deduced from these, will satisfy the original equation, and the same values, when substituted for x and y in any other equation, will produce an equation, from which may be found, thence x' and y', also x and y may be found.

4. Equations of the third order, involving two unknown quantities, may, in certain cases, be satisfied by indeterminate functions of x and y; when such an equation is combined with any other equation, the method here explained will apply.

5. By the analytical artifice here explained, equations which involve functions of unknown angles may be transformed into common algebraic equations.

For example, let it be required to determine the angles and from the equations,

a tan. +b tan. 4-c........... (2)

In addition to these, the arithmetic of sines furnishes the two equations,

2

2

Cos. + sin.q=1, cos. + sin. 1,

which are satisfied by making

The values of r and s may now be found by the usual

methods.

ROYAL MILITARY COLLEGE,

Feb. 4. 1819.

ART. IV.-Description of a New Hygrometer, made of the internal membrane of the Arundo Phragmites*. By ALEXANDER ADIE, F. R. S. Edin. Communicated by the Author.

IN the winter of 1816, I made many trials of different substances, for the purpose of ascertaining their hygrometric powers, in order, if possible, to find one which should possess sufficient sensibility, and, at the same time, not be liable to change the extent of its contraction between the extremes of dryness and humidity. Among the various substances which I tried, those that changed their bulk in a considerable degree by a change of humidity were Rottenstone, Chalk, unbaked Clay made very thin, and Mountain Cork. Though, from the friable nature of the three first, it was found difficult to use them, yet I am of opinion that they may be advantageously employed in the construction of hygrometers, and there is reason to think that they will not be subject to any alteration in their scales.

Charcoal, from its known durability, likewise presented itself as a proper substance for the above purpose; and it was found, upon trial, to be sensibly hygrometric, although its range was very limited.

Most of the above substances were formed into hollow cylinders, and cemented to the end of thermometer tubes; and their expansibility was tried by filling the cylinder and tube with mercury, in the usual manner.

But the substance which was found to possess by far the most delicate sensibility, and extensive range, was the internal membrane of the Arundo Phragmites. A small bag, made of this membrane, is attached to the lower end of a thermometer tube, so as to form, as it were, its bulb. It is then nearly filled with quicksilver, which rises and falls, in consequence of the contraction and dilatation of the membrane, by any change of moisture; and these changes are indicated upon a scale attached to the tube, the zero of this scale marking absolute humidity, and the other extremity of the scale absolute dry

* This Paper was read before the Royal Society of Edinburgh, on the 1st of February 1819.

ness.

The lower end of the glass tube, instead of being merely inserted into the top of the bag, may pass through it, the quicksilver in the bag communicating with that in the tube by one or more openings made through the sides of the tube. By this means, the bag is supported by the glass, and prevented from being injured by any slight accident; and the instrument is also less affected by any change of temperature,

A convenient portable hygrometer may be made, by employing a slip of this membrane, and attaching its extremities to the end of a lever, somewhat like the small pocket metallic thermometers. The external appearance of one of these instruments is shewn at the bottom of the Patent Sympiesometer, represented in Plate II. Fig. 2.

Although this membrane is not entirely free from the change to which all animal and vegetable substances are liable, yet hygrometers made of it possess a considerable degree of uniformity amongst themselves; and, in point of sensibility, this membrane exceeds every other substance that I have met with.

ART. V.-Account of an Excursion to Thebes, and of the Antiquities recently discovered in that City. In a Letter from a Scotch Gentleman in Cairo.

CAIRO, August 11. 1818. FINDING that I was to be detained here for some time, you will not be surprised that I undertook an excursion to "the city of a hundred gates." I considered a sight of its temples, &c. as forming an era in a person's life; and a more favourable opportunity for visiting them, could not possibly occur. As soon as I had determined upon the journey, I was favoured by Mr Salt with letters for several people established near my route, and having a Firman from the Pacha, which I afterwards found was quite unnecessary, as the peasantry are every where anxious to serve and to oblige, a canja, or pleasure-boat, was procured, which cost us little more than a palanquin hire in India. We left Cairo on the 25th June 1818, and made very rapid pro VOL. I. NO. I. JUNE 1819.