Philosophical Transactions of the Royal Society of London

either theory. According to the one which has been commonly received, the oxygen unites with the real acid of muriatic gas, which becoming oxymuriatic acid, deposits water. On Sir H. DAVY's view, the oxygen unites with the hydrogen of the muriatic acid, and composes water, while the oxymuriatic acid is merely an educt. I am not aware of any refinement of the process, by which the value of these two explanations can be compared. Something, however, would be gained by a precise determination of the proportions, in which the two gases saturate each other. For since, on Sir H. DAVY's theory, muriatic acid contains half its volume of hydrogen gas, two measures of which are known to be saturated by one of oxygen, it follows that muriatic acid gas should be changed into oxymuriatic by one-fourth of its bulk of oxygen. According to GAY LUSSAC and THENARD,* three measures of muriatic acid should condense one of oxygen (or only one-third their bulk), and should form two measures of oxymuriatic acid. Hitherto, I have not been able to satisfy myself respecting the true proportions of oxygen and muriatic acid gases, that are capable of being united by electricity; for though I have made several experiments with this view, they have not agreed in yielding similar results. The condensation of a part of the undecomposed acid by the water, which is formed during the process, will, probably, indeed, always be an impediment to our learning these proportions exactly. The fact is chiefly of value, as it affords an example of the production of oxymuriatic acid under the simplest possible circumstances; and as it shews unequivocally that, under such circumstances, the visible appearance of moisture is a part of the phenomena.

• Mémoires d'Arcueil, ii. 217.

Manchester, Jan. 6, 1812.

XVI. Of the Attraction of such Solids as are terminated by Planes; and of Solids of greatest Attraction. By Thomas Knight, Esq. Communicated by Sir H. Davy, LL. D. Sec. R. S.

Read March 19, 1812.

MATHEMATICIANS, in treating of the attraction of bodies, have confined their attention, almost entirely, to those solids which are bounded by continuous curve surfaces; and Mr. PLAYFAIR, if I do not mistake, is the only writer, who has given any example of that kind of inquiry, which is the chief object of the present paper. This learned mathematician has found expressions* for the action of a parallelopiped; and of an isosceles pyramid, with a rectangular base, on a point at its vertex; and observes, on occasion of the first mentioned problem, that what he has there done, " gives some hopes of

being able to determine generally the attraction of solids "bounded by any planes whatever.”

It is this general problem, that I venture to attempt the solution of, in what follows: viz. any solid, regular or irregular, terminated by plane surfaces, being given, to find, both in quantity and direction, its action, on a point, given in position, either within or without it.

* Ed. Trans. Vol. VI. p. 228 to 243. It is proper however to observe, that Mr. PLAYFAIR'S expression, at p. 242, for the action of a parallelopiped, requires to have its sign changed; being, as it stands at present, negative, from the manner of correcting the fluent.

Nor has the matter any difficulty, as far as theory* only is concerned; although, actually to find the attraction, of a body of very complicated figure, may, no doubt, be exceedingly laborious and troublesome: for no one, I suppose, will conceive, that it can be done in any other manner, than by a previous partition into more simple forms, each of which must have its action found separately.

Having completed this part of my subject in the three first sections, I next apply the formulas; given in §. 1, to find the attraction of certain complex bodies, which, though not bounded by planes, have yet a natural connexion with the preceding part of the paper. Finally, the fifth section treats, pretty fully, of solids of greatest attraction, under various circumstances; and I do not know, that any one of the problems there given has been before considered by mathematicians; whilst, on the other hand, the results of former writers are easily derived as corollaries.

For the sake of perspicuity, I have divided the paper into propositions, and shall terminate this short introduction by expressing a hope, that I may not be chargeable with unnecessary prolixity.

§. I.

Of the Attraction of Planes bounded by right Lines. As all such figures may be divided into triangles, it seems natural to begin with these.

* It is usual, I think, with mathematicians, to consider a thing as done, when it can be pointed out how it may be done. Thus M. LAGRANGE, in his excellent work "De la Résolution des Equations numériques,” says (p. 43) " cette méthode ne laisse, " ce me semble, rien à desirer." where, of course, he can only mean, as far as relates to theory.

Prop. 1.

Let rvm, fig. 1, be a triangle, right angled at r, and pm a right line, perpendicular to the plane of the triangle, at the angular point m; it is required to find the attraction of the triangle, on the point p, both in quantity and direction.

•

Conceive a plane to pass through the point p, parallel to the plane of the triangle, and, in it, the lines pg, po, respectively parallel to rm, rv. The problem will be solved, if we find the actions of the triangle, in the directions of the three rectangular co-ordinates pm, pg, po.

Draw ks parallel to rv, and put a = pm, b=rm, T=mk, t=kq; then pq = √ aˆ+T2+t. Let r= tang. vmr, ks=rx km.

then

The element of the plane at q is İx t, and its action, on p,

ři

in the direction pq, is a2+T2+¿2; by resolving which, and put

ting A, B, C for the actions of the triangle, in the directions pm, pg, po, we get

A=ff

ari

; B=SS.

Tri İti ; C=SS (a2+T+t2) š (a2+P2+t2)ž (a2+T2+12)} in all which expressions, we must first take the fluent, with respect to t, from to, to trT; and afterwards, with respect to T, from T= o, to Tb. To begin with A,- a first operation gives

arTi A =S; (a2+T2) (a2+(1+r2)T2){ which, if we put B =1+r2, will be changed to A =S —_____ar TF

B (a2 + T2) (~/+T2