a notion. On the other hand, it is rather remarkable that Professor Young did not consider that was the usual symbol of absurdity or of incompatible conditions, and that could never be so, in the result of an investigation logically conducted. Thus, the corresponding antecedent equation to the result =o, when cleared of fractions, is ox = 0 or 0 = 0, an equation that is very obviously satisfied without any limitation to the value of x, and that cannot fail therefore to be compatible with other equations or conditions; but the corresponding antecedent equation to the result is o = Q, an equation evidently indicating the presence of absurdity or of incompatible conditions, unless the nature of the investigation will admit of infinite results. The query respecting the geometrical series is dismissed at once by a reference to the fourth extract from my essay. By putting for S the series it represents, the equation is n-1 a + ar + a r2 + ...... + ar = a (r" — 1) 1 and as the left-hand member is not discontinuous when r = 1, the vanishing fraction, which forms the right-hand member, must be limited to its continuous value, viz. na. The very circumstance of the equation involving both a determinate and an indeterminate quantity, when r=1, indicates the existence of a fallacy in the process by which it has been deduced. We first have (r− 1) Sarn — a = a (pTM” — 1) ................ (b) which divided by r — 1, gives - (c) In the case = 1, and r - 1 o, we have therefore committed the fault of multiplying by absolute nought in passing from (a) to (b); but the equation (c) is a true deduction from (b), for the mere placing of r-1 in the denominator of a fraction is not an actual performance of division. The equation (a) becomes S = a + a + a + ......; the equation (b) entirely vanishes, and (c) becomes S = After the foregoing discussion it will be needless to offer any special observations on the obvious inaccuracy of Professor Young's views of the ellipse question. It may, however, be worth while to take the opportunity of adding a single remark on an erroneous principle which he appears to entertain regarding the general theory of analytical results. I never before heard of the incompetency of an analytical result to afford any positive information that an investigation could admit of. It is plain that the original equations, which express the analytical conditions of a problem, cannot include any extraneous conditions with those expressed in the enunciation, and that they must therefore comprehend, in their analytical results, every solution that the problem is capable of receiving. The equations, however, may not include certain other implied conditions, dependent on the peculiar nature of the inquiry, and therefore may yield some additional solutions incompatible with the conditions so implied. For instance the nature of a problem may be such as to exclude from the results not only imaginary values but negative values and values which fall beyond certain limits, though they will be unavoidably comprehended in the analytical solution. The exclusion of inadmissible solutions, therefore, rests with the nature of the problem and not with the forms of its analytical conditions. It is hence evident that Professor Young involves himself in a palpable inconsistency, when he arrives at the fact of the ellipse question admitting multiple solutions, by an examination of the original analytical conditions, and at the same time alleges that the analytical result is quite incompetent to supply that information; for the true analytical result must necessarily present every solution capable of satisfying the analytical conditions from which it has been deduced. If we refer back to the nature of the problem, as originally presented, which is the proper source of rejective information, we perceive that the only condition it imposes on the results is the limitation which requires the coordinates xy to fall within the bounds of the ellipse, or of the circle that represents it in the indeterminate case. I have thus unreservedly enumerated the principal reasons on which I found my sincere and firm conviction of the incorrectness of the various statements contained in Professor Young's letter. To avoid the possibility of being misunderstood, I have also given a concise analysis of the most important of the principles maintained in my essay; and, in conclusion, I may be permitted to add, that instead of their being "condemnatory of conclusions which, in the works of our ablest modern analysts, wear all the aspect of mathematical certainty," they establish the truth of those very conclusions on a firmer and more intelligible basis,—that instead of the mere aspect of certainty, in favour of those conclusions, they substitute certainty itself. London, April 9, 1836. LXXII. Further Experiments on Conducting Power for Elec* tricity. By EDWARD SOLLY, Jun., Esq. 15. IN my former communication I said that I had found iodine when solid to be a nonconductor, but I did not describe any experiments made with it in the melted state. This perhaps may have appeared an omission, the more so after Dr. Inglis's note (the contents of which had not, however, been communicated to me,) had been appended to my paper; but I had been advised to lose no time in describing such of my experiments as were in opposition to Dr. Inglis's statement that iodine is a conductor ".+ What follows now will explain that apparent omission. 16. In all my original experiments I had found iodine to be a nonconductor in the fluid as well as the solid state; but on the present occasion, when I was led to repeat them by the abovementioned statement, I was not a little surprised to find the iodine, when rendered fluid by heat, become a conductor. That a substance acting as iodine does should not be similar as to conducting power when fluid to what it is when solid, (as all known substances that have been as yet examined are, excepting only such as are electrolytes, and also perhaps the periodide of mercury,) but should appear a conductor upon assuming the liquid state, was so singular, and so contrary to my previous results and preconceived views, that I was induced to multiply my experiments; they continued unsatisfactory, and they were the more so as the iodine did not always appear a conductor but sometimes a nonconductor, and then, when it did appear a conductor it did in a very feeble manner, and with great uncertainty. 17. I was therefore led to doubt the purity of the iodine which I was using, and this seemed the more probable as it was from a different source from that which I had employed in the original experiments; and in the means which Ì had before described for examining conducting power it was impossible the wires could touch, and therefore the objection which the use of loose wines would have introduced was avoided. In consequence I procured some perfectly pure iodine sublimed at a very low temperature, and ascertained that that which I had Communicated by the Author: see our Number for February, p. 130. ↑ Lond. and Edin. Phil. Mag., No. 43. p. 129. been previously using contained some impurity, most probably the iodide of iron, which is not unfrequently present in the iodine of the shops. The pure substance which I now used proved equally a nonconductor when fused as it had proved to be when solid. 18. When iodine is distilled with five times its weight of chlorate of potassa, a liquid comes over, which, according to Wöhler, is a chloride of iodine: it proved to be a very good conductor. Its solution in æther was also a good conductor, æther being as is well known a non-conductor. The chloride which I used was purified by being twice distilled off chloride of calcium. After the electric current had passed, upon examining the tube which had con-tained the chloride of iodine, I found that the one platinum wire, or that which had been the anode, was very much corroded, but still quite clean; the other, or that which had been the cathode, was encrusted with black matter very like iodine in appearance. So good a conductor indeed was this fluid, that the spark of a voltaic battery was hardly visibly impaired by interposing a small portion of it in the circuit. Great heat was evolved during the passage of the current, so that the liquid soon boiled. 19. The chloride of bromine and its solutions in water and æther were all good conductors. 20. I prepared iodic acid by Connel's process and then heated it up to its boiling point. I kept it fused and boiling for about a minute, and then allowed it to cool; by this means more than half was decomposed and volatilized, but what remained was I believe pure iodic acid. I used it immediately after this to prevent absorption of moisture from the atmosphere. I then found it a most distinct insulator when solid, but a very good conductor when fused, so much so that a spark might be easily taken from its melted surface. Its aqueous solution was also a very good conductor, and when strong, iodine was precipitated at the cathode. 21. It is very interesting and curious that iodic acid should behave thus, for as in all hitherto described experiments oxygen and iodine were both found to go to the same electrode, and as in order to the decomposition of a body the two composing ïons must go to opposite electrodes*, it seems very unlikely that iodic acid should be an electrolyte: besides this, it is not composed of one proportional of each of its elements, which Mr. Faraday has shown to be the case with all known electrolytest. Experimental Researches in Electricity, by Mr. Faraday, No. 828.[Lond. and Edinb. Phil. Mag., vol. v. p. 425.-EDIT.] + Ibid. No. 679.—[vol. v. p. 167.] Third Series. Vol. 8. No. 48. May 1836. 2 T If, however, it be an electrolyte, which is very improbable, it will be a proof that in the electrolyzation of a substance the evolution of the one ïon depends entirely on the nature of the other ïon with which it is combined; and thus the terms anïons and catïons will only be relative. If, however, iodic acid is not electrolyzed, still this experiment furnishes us with another exception to the law of liquido-conduction similar to the periodide of mercury†. * 22. Unfortunately, however, iodic acid is decomposed by the same degree of heat which is required to melt it, and the vapours of iodine entirely prevent the acid being examined during the experiment: it is also decomposed by almost all substances which can be used as electrodes, and therefore the advantage which can sometimes be taken of observing which of the electrodes is corroded, is here of no avail. I was therefore quite unable to ascertain whether iodic acid was electrolyzed or not; but when the electrodes were immersed in the fused acid, much stronger ebullition seemed to take place than before. 23. It was impossible to ascertain whether the oxides of bromine and chlorine were conductors or not, and I therefore had not the advantage of comparing iodic acid with the bromic and chloric acids. I had at first some hopes of being able to add further experiments in relation to these last described, but finding that not in my power, I no longer delay sending the above. 7, Curzon Steeet, 15th April, 1836. LXXIII. Reviews, and Notices respecting New Books. On the Theory and Solution of Algebraic Equations; with the Recent Researches of Budan, Fourier, and Sturm on the Separation of the Real from the Imaginary Roots of Equations: by J. R. YOUNG, Professor of Mathematics in Belfast College. Souter, London. WE WE have more than once dwelt upon the remarkable perspicuity of Mr. Young's writings. In this respect they are, one and all, models of the very best kind for the elementary writer, and far better adapted than any which we are acquainted with, for the purposes of actual study. In saying this, we mean no ordinary praise; for of all kinds of writing on science, and especially on mathematical science, the development of elementary principles in a perspicuous and logical manner is the most difficult. If Mr. Young had succeeded only in this, beyond any other author in our language, he would have achieved much, and have effected sufficient towards a diffusion, not only of Exp. Res. in Electricity, by Mr. Faraday, No. 402.-[Lond. and Edinb Phil. Mag., vol. iii.] +Ibid. No. 691.-[vol. v. p. 169.1 |