represents the root of another condition that has been improperly derived from it.

At page 516, Prof. Young states that I have only discussed the converse of the Proposition III. in my former letter. This, however, is not the case, for in that letter I distinctly go into the demonstration of each of the four propositions extracted from my essay. Indeed, immediately after, at page 518, Professor Young himself admits that I have done so !! In allusion to my having rejected as illogical the processes in which multiplications or divisions by zero occur, he there states that "if only results obtained under such restrictions as these are admitted to come under the second and third principles, then the generality of those principles is, of course, at once given up, and my friend and I are thus far agreed." Now these restrictions are no more than the necessary exclusion of results obtained by imperfect and fallacious reasoning, and cannot fairly be considered as limitations to the generality of the propositions. It is now distinctly avowed by Prof. Young that the generality of the propositions objected to is established for all cases in which these restrictions are attended to, or in which the fallacious reasoning does not enter. Proceeding on this admission, it necessarily follows that in every case in which the final result fails to conform with the propositions the fallacious reasoning must have been introduced somewhere in the process of solution. In every possible case, therefore, of nonconformity with the Proposition II. or III. we observe that the fact itself is a sure indication that a multiplication or division by zero has occurred in the investigation, and therefore that the case must be rejected at once as not offering a legitimate result. It would be utterly useless, therefore, to enter into Prof. Young's observations at page 517, on the expression for the radius of curvature*, since neither this nor any other particular example of nonconformity can have the least weight unless my friend can show the operations of multiplying and dividing by zero to be strictly logical.

Prof. Young is quite out in supposing that the restrictions to correct reasoning will limit the application of the principles

• I would, however, here remark that the conditions P = 0, Q = 0, will not necessarily cause a value of d r to be zero, as Prof. Young alleges. Р Such a doctrine would imply that any vanishing fraction

Q

would neces

sarily attain its greatest or least value when its numerator and denominator both vanish, which is not the case. The conditions P = 0, Q = 0 will not, therefore, determine the points of higher contact referred to by Professor Young. In his example of the parabola the determination of the true result is purely accidental.

1.

to

comparatively few cases." On the contrary, the false cases very seldom occur, and they can generally be corrected by a slight reference to Proposition IV. This is, in fact, the very way in which I dismiss Prof. Young's query respecting the sum of the geometrical series. I refer to Proposition IV. not to interpret but to correct the resulting expression for the particular case in which the reasoning has failed. Had the expression been a just deduction, Proposition III. would have applied to it, and the sum in that case would have been anything. How Professor Young could have misconceived me I am quite at a loss to explain, as I have distinctly pointed out the fault that occurs in deducing it.

In Prof. Young's first letter, at page 298, he states that "when we are operating with equations of the first degree containing several unknown quantities, the symbol is, in fact, the very form which the result usually takes when the proposed equations involve incompatible conditions." I feel assured that Prof. Young, after a little reflection, will not venture again to assert the truth of this hasty and erroneous statement. At page 520, however, he has attempted to refute my observation that can never be the symbol of absurdity in the result of an investigation logically conducted; but it will be seen that my friend's remarks are founded on the same erroneous hypothesis, that the result is justifiable in whatever way it may have been deduced. When it is the result of a logical process it is obvious, since the antecedent equation is satisfied by any value, that all the preceding equations, and consequently the original condition, must likewise be satisfied by any value. The symbol, whenever it is the result of a strict investigation, may therefore possess any value, and cannot possibly be the "symbol of absurdity," however much Prof. Young may be "surprised" at the statement.

In thus candidly replying to my friend's letter, I am disposed to give him every credit for his own opinions. I think, however, that he ought not, for his own sake, to have proceeded to such a length with his remarks, without having, in -the first place, made some attempt in support of his objectionable premises. Should Prof. Young still entertain the same opinions I shall make no further attempt to change them, though I may be allowed to maintain my own. It will not be necessary to enter into any further details at present, as I have doubtless said quite sufficient for the mathematical readers of the Journal. At least I am well satisfied with having shown, Third Series. Vol. 9. No. 51. July 1836. E

by concise, general, and undeniable reasoning, that my friend might have advantageously spared himself the trouble of offering his numerous observations, had he paid more regard to the logical strictness of his assumptions.

London, June 4, 1836.

By

VII. On the Water of Crystallization of Soda-alum. THOMAS GRAHAM, F.R.S. E., Professor of Chemistry in the Andersonian University, Glasgow; Corr. Member of the Royal Academy of Sciences of Berlin, &c.*

THE

HE double sulphate of alumina and soda crystallizes in the form of the regular octohedron, like the sulphate of alumina and potash, while the former salt is supposed to contain twenty-six atoms of water, and the latter contains only twenty-four. The coincidence in form of these two salts is most interesting, for in no other corresponding salts of potash and soda has such a relation been observed from which any inference in respect to isomorphism could properly be drawn. Yet if the soda-salt contains two atoms more of water than the potash-salt, the conclusion which follows is, not that soda and potash are isomorphous bodies, but that soda plus two atoms water is isomorphous with potash, as ammonia plus one atom of water is isomorphous with the same body. But the last analogy is superficial and likely to prove illusory.

The exact determination of the water of crystallization of a salt is often a problem of no inconsiderable difficulty, as many precautions must be taken which are by no means obvious. To have alumina free from potash or ammonia, it was precipitated from pure potash-alum by means of carbonate of soda. A solution of sulphate of alumina was formed by dissolving the precipitated alumina in the proper quantity of sulphuric acid, and the requisite proportion of sulphate of soda was added. A considerable crop of crystals of soda-alum were obtained from the above solution allowed to evaporate spontaneously in air.

Like most very soluble salts the crystals of soda-alum, when newly prepared, retain hygrometrically a notable quantity of the saline liquor in which they have been formed. But the crystals of this salt cannot be dried easily, as after they lose their hygrometric water they are nearly as efflorescent as sulphate of soda itself. Before being submitted to analysis the crystals had been kept for five months of cold weather in a large phial stopt by a cork. Their surfaces remained per

* Communicated by the Author.

fectly bright, and not in the slightest degree effloresced; but the crystals had lost, from the escape of their hygrometric water, that extreme and watery clearness which we have in the crystal newly removed from its mother-liquor. From my experience in respect to such salts, I had reason to believe that the crystals of soda-alum were now in a most suitable condition for analysis.

Upon a very damp day the crystals were reduced to powder and pressed in blotting-paper. A large crystal exposed to the air at the same time lost nothing. 20-35 grs. of the salt so prepared were exposed on a sand-bath to a heat, which was gradually raised so as to effloresce the salt without melting it or causing vesicular swelling. In eight hours the salt had been heated above the melting point of tin, and had lost 8.98 grains. It was thereafter heated, in a gradual and cautious manner, to low redness by the spirit-lamp, and the loss became 9.65 grains. By a continued exposure to the same heat for half an hour more, the salt lost only one hundredth of a grain additional. Supposing it now to have lost all its water, the salt will consist of

The calcined salt dissolved slowly but completely in boiling water. By precipitation with muriate of barytes it afforded 21.22 grains sulphate of barytes, equivalent to 7.37 grains sulphuric acid. Or the crystallized salt contains 34.73 per cent. of sulphuric acid, while the theory of twenty-four atoms of water supposes it to contain 34.93 per cent. sulphuric acid. It follows from this analysis that soda-alum contains twentyfour atoms of water and not twenty-six.

There is no reason to question the perfect accuracy of the analysis of potash-alum by Berzelius, which gives to it likewise twenty-four atoms of water. Dried in the manner described for soda-alum, I found it to consist of

In such analyses, there is imminent danger of the water carrying off a little acid with it, unless it is expelled in the

most slow and cautious manner. It is probably from this

cause that the water has come to be overestimated in the case of the alums. But they stand a low red heat without decomposition, if first made quite anhydrous.

VIII. Second Theorem of Algebraic Elimination, connected with the Question of the Possibility of resolving, in finite Terms, Equations of the Fifth Degree. By Professor Sir WILLIAM ROWAN HAMILTON, Astronomer Royal of Ireland.

(In continuation of a Communication in the last volume, p. 538.)

Theorem II. IF x be eliminated between a proposed equation of the fifth degree,

x5+Ax4+Bx3 + C x2 + D x + E = 0,

and an assumed equation, of the form

y = Qx+f(x),

in which f(x) denotes any rational function of x,

(55.)

(2.)

(3.)

and if the constants of this function be such as to reduce the result of the elimination to the form

y+By+D'y+ E' = 0, ...............

independently of Q: then not only must we have

(56.)

(57.)

so that the proposed equation of the fifth degree must be of

the form

25+ B3 + D x + E = 0,

.....

but also the function f(x) must be of the form

(58.)

f(x) = qx+(x2 + B x3 + D x + E). (x), ... (59.) q being some constant multiplier, and (x) some rational function of x, which does not contain the polynome 25+ Ba3 +D+ E as one of the factors of its denominator; unless we have either, first,

E = 0,

(60.)

or else, secondly,

5 D = B2,

(61.)

or, as the third and only remaining case of exception, 55 E4+22 B E (2° 53 D2-32 52 B2 D+ 33 B4)

+2* D3 (24 D2-23 B2 D+ B1) = 0.

(62.)