gest the real values of the integrals; for example, write bi (i=√/−1 as usual) for b in

0

The former result is well known; and the latter is easily verified by differentiating with respect to b and forming a differential equation, from which the value of the integral can be determined. As another example, writing bi for b in (14), and noticing that

e-cr2 sin 2bx dx=0,

Eef (a√e+ b) + Eef (ave- bir) -2 Erfave

a result which can be verified by forming the differential equation

y denoting the integral, which, on integrating and determining the constants by the considerations that the result must be independent of the sign of b, and that when b=0 it must equal

gives the same expression for y.

Erf (ac),

The result (25) may be written in another form for

XXXIII. On a new Method of solving some Problems in the Calculus of Variations, in reply to Professor Cayley. By the Rev. Professor CHALLIS, M.A., F.R.S.*

N an article in the Number of the Philosophical Magazine

september, Cayley

from the new method of solving certain problems in the Calculus of Variations which is contained in my communication to the Number for July. On carefully considering all that he has

* Communicated by the Author.

said, I find that no argument is adduced which does not rest on the assumption that the equations Ap=0 and A=0 are necessarily identical. To this assumption, which is made without the support of any reasoning, I oppose the general argument, that as the two equations differ symbolically (the former being of a degree superior by one to that of the other), they cannot be necessarily equivalent, inasmuch as in that case the symbolic difference between them would have no signification, which is contradictory to the principles of analysis. This à priori reason I proceed to confirm by the following particular considerations.

There are instances in which Ap=0 and A=0 are both integrable per se, and the two integrals are identical. The Problem I., solved in my article in the July Number, presents one such instance. In other cases, as that of the problem proposed by Mr. Todhunter in p. 410 of his History of the Calculus of Variations,' only Ap=0 is integrable per se, but the integral satisfies A=0. In these two classes of solutions the Calculus gives for each problem a unique result.

There are also instances in which a solution is effected by an integral of Ap=0 which does not satisfy A=0. This is the case with respect to the problem of the greatest solid of revolution of given superficial area, the surface being subject to the condition of passing through two given points of the axis. The discontinuous solution of this problem obtained by Mr. Airy in the Number of the Philosophical Magazine for July 1861, is deduced exclusively from the equation Ap=0. In the Number for June 1866 Mr. Todhunter has completed this solution by proving that it actually gives a maximum; but in so doing he has expressly excluded the equation A=0. In short, in this instance the equation Ap=0 is treated as if it were independent of the equation A=0. I am entitled, I believe, to say that Professor Cayley assents, as I do, to the solution in question. Why, then, does he object to treating A=0 independently of Ap=0? The result which he accepts proves that the two equations are not necessarily equivalent.

In the July Number I have deduced solutions of Problems II. and III. by an independent treatment of the equation A=0, and have thus obtained, in the case of Problem III., the continuous solution of the problem of which, as stated above, the Astronomer Royal gave a discontinuous solution. The novelty of the process I have adopted consists in deriving from the equation A=0, regarded as the differential equation of a curve, the equation of the corresponding evolute, and then employing one of the involutes to satisfy the conditions of the proposed problem. In the case of Problem II. the equation of the evolute was explicitly obtained, and the appropriate involute could consequently

be immediately determined. With reference to this point, Professor Cayley says, "after the evolute is obtained, we must take not any involute, but the proper involute of such evolute." But, as Professor Cayley well knows, there is no proper involute of any evolute, because the number of the involutes of a given evolute is unlimited. What he calls "the proper involute" is the curve given by integrating the equation Ap=0; and on the before-mentioned gratuitous assumption that this equation and A=0 are identical, he refuses to recognize any involute derivable from the latter equation other than that curve. Of course I do not admit that this is an argument, because, for the reasons already urged, I maintain that there is no ground for the initial assumption.

The equation of the involute which gives the solution of Problem II. contains three arbitrary constants, because it involves the arbitrary length of the cord which, by unwinding from the evolute, describes that involute. By eliminating the three constants a differential equation of the third order is obtained, which is evidently not identical with the equation A=0 of the second order. Neither is it identical with =0, because it cannot be

dA
dx

satisfied if A a constant. But it is found that that equation

day day
dx2

of the third order is verified by substituting for and

dxs

the

values of these differential coefficients deduced from A = 0

dA dx

and its derived equation =0. This is proof that the involute

which may be regarded as the solution of the problem is strictly derived from, and exclusively depends upon, the equation A=0. The process of derivation by the intervention of an evolute I have called "a new integration," as being distinct from the mode of solution by ordinary integration.

The mathematical reasoning with which Professor Cayley concludes his communication only amounts to a proof that the integral of Ap=0 gives a curve which is included among the involutes of the evolute which was derived from the equation A=0. I have no remark to make on this result, as I had already obtained the same in the article in the July Number.

For the reasons above alleged, I adhere to the statement made at the end of the former communication, namely that I have succeeded in removing from analytics the reproach of failing to solve certain problems in the Calculus of Variations.

Cambridge, September 6, 1871.

XXXIV. Contributions to the History of the Phosphorus Chlorides. By T. E. THORPE, Ph.D., F.R.S.E.*

IN

I. On the Reduction of Phosphoryl Trichloride.

N his first memoir on Vanadium, Dr. Roscoe described a series of oxychlorides obtained from vanadyl trichloride by the action of reducing-agents. When the vapour of vanadyl trichloride is passed together with hydrogen through a heated tube, a bright grass-green crystalline sublimate of vanadyl dichloride, VO C12, is produced in the anterior portion of the tube; afterwards a layer of vanadyl monochloride, VO Cl, is deposited as an exceedingly light, flocculent, brown powder; whilst at the extreme end of the tube beautiful bronze-coloured plates of the divanadyl monochloride, V2 O2 Cl, are formed, which have the appearance of mosaic gold. In this memoir Dr. Roscoe clearly pointed out the intimate analogy which exists between the compounds of vanadium and those of phosphorus, arsenic, antimony, and nitrogen; and in his subsequent researches on this subject, he has so far elaborated this view of its chemical relationship, that there is no longer room to doubt that vanadium is virtually a member of the trivalent group of elements.

It must be confessed, however, that the triatomic nature of vanadium is not very apparent in the oxychlorides derived from the vanadyl trichloride if the simplest formulæ derived from their analysis are retained; but if these formulæ be doubled, the difficulty at once vanishes. The supposition that these oxychlorides possess a greater molecular weight than the vanadyl trichloride, may derive some support from the fact of the change of physical state which accompanies their formation, the lower oxychlorides being all solid. Beyond this I am not aware that any fact is known to establish such an assumption, unless it be the coincidence between the atomic volume of the vanadyl trichloride and that of the vanadyl dichloride with the formula doubled.

According to this view, the formulæ of these oxychlorides and their relation to the vanadyl trichloride would be graphically represented thus:

Communicated by the Author, having been read at the Meeting of the British Association at Edinburgh, September 1871.

Phil. Mag. S. 4. Vol. 42. No. 280. Oct. 1871

X