Hence for the ratio of the radii vectores from the two poles for the points P, Q, R, S, we have respectively; that is, we have for all these points the common ratio Mn-m. Similarly, we should find for the points P, Q1, and R, the common ratio un-m+1, and for the points Q', R', and S' the common ratio un-m-1. Hence, not only are the equipotential lines for two equal and opposite poles charaeterized by a constant ratio of the radii vectores, as already proved (§ 16), but this ratio changes in a constant ratio on passing from any one line of the system to the next, the ratio of change (=u) being the same as the ratio of change of radius on passing from line of the system due to a single pole to the next. 19. The actual potential at any point of the sheet, in terms of the distances of this point from the two poles, follows directly from equation (2) in $ 7. Letr be the distance of the given point from the source, and ow its distance from the sink; put V for the potential at the point due to the source alone, and V' for that due to the sink alone. Then we have Q 2πκδ and Q' log m; 2пко and since the source and siuk are equal, Vi=-V, and Qz-Q; therefore the resultant potential, or V+V', is Q g U= log (4) 2πκδ This gives the potential =0 at all points of the straight line equidistant from the two poles, positive on the side of this line next the source, and negative on the side next the sink. Also it shows that for equal differences of potential we must have equal differences in the value of logs, which agrees with what was proved above ($ 18). logo . . We Q 2πκδ whence, regarding the source and sink for the present as mere points, the value n=will correspond to the former, and n=-0 to the latter, while n=0 denotes the straight line at equal distances from both. 20. Let the distances from 0 (the middle point of A B) of the points D and D' (fig. 2), in which the circle of potential U cuts A B, be called l and l'; then, a being as usual half the distance between the poles, ati la =u", where hen is the ratio of the radii vectores of the given circle. Hence Il=a', and l=a MP + 1 l'=a u" -1° For the radius of the circle, we have evidently 2au" p=i(1-1)= uan-1' u2n-1 and for the distance of its centre from 0, d=l+p=l-p=a 42n +1 ; also p?=CA.CB=(d+a)h=do — a?, where h(=d-a) is the distance of the centre of the equipotential circle from the nearest pole. 21. In order to draw a system of equipotential circles with compasses, it is most convenient to have given the distances of the centres from the nearest pole, and also the distances from the same point of one of the intersections with A B, the line joining the poles—that is, in fig. 2 the distance B C and either BD or BD'. Now 2a BC=h=d-a= 2n-1' M2n. 2a BD=a-l= un +1' 2 E 2 if a a+L m a Any convenient value such as 2 or may be given to u; or special number of lines be required, a value may be found to suit. Thus let L be the greatest value that is to be given to 1, and m the number of lines required on each side of O. We shall have a+L 1 a-L=ht, or log u= log -L' which determines the value of M. The successive values of B D' are then 2a 2a 2a 2a — і р? — 3 — 1 The successive values of BC are simply alternate values of these, namely 2a 2a 2a M2_1' 14-1' Plate IX. shows the equipotential lines drawn for a value of un-1 C . u2m ,2m 1 [To be continued.] XLV. Notices respecting New Books. The Principles of Science : a Treatise on Logic and Scientific Method. By W. STANLEY JEVONS, M.A., F.R.S., Fellow of University College, London ; Professor of Logic and Political Economy in the Owens College, Manchester, London: Macmillan and Co. 1874 (two vols. Svo, pp. 463 and 480). THIS HIS is in many respects a remarkable book, and particularly so in regard to the extraordinary number, variety, and (as far as we can venture to judge) accuracy of the facts which are brought in illustration of the principles and questions discussed. There is indeed scarcely a branch of science which the author has not laid under contribution for purposes of illustration. The work is divided into five books, viz. :-On Formal Logic, Deductive and Inductive; on Number, Variety, and Probability; on Methods of Measurement; on Inductive Investigation; and on Generalization, Analogy, and Classification. We have found it very difficult to state in few words the upshot of the discussions in these books severally without doing great injustice to their contents. For instance, the first book might be said to be an exposition of formal logic designed to set forth the author's peculiar method of expressing propositions and arguments by symbols, and to lead up to an account of his logical machine-a machine worked by twenty-one keys like a pianoforte, that « and able to draw inferences, or, at least, “to represent a mind endowed with powers of thought but wholly devoid of knowledge" (vol. i. p. 127). This, however, would be to give a very inadequate notion of the contents of this first part, though we do not see how to give a better account of it in few words. So, again, the second part might be said to contain an exposition of the doctrines of permutations and combinations, and of probabilities, which is closely connected with them, designed to lead up to the position we cannot adequately understand the difficulties which beset us in certain branches of science, unless we gain a clear idea of the vast number of combinations or permutations which may be possible under certain conditions. Thus only can we learn how hopeless it would be to attempt to treat nature in detail, and exhaust the whole number of events that might arise” (vol. i. p. 216). Yet this is not an adequate statement of the aim of Book II. ; and the same would be found true of similar statements made in regard to the other books. Of course, in the case of any elaborate treatise the same may to some extent be true ; but in the work before us the difficulty assumes unusually large dimensions, and renders the task of the reviewer peculiarly hard. Probably the merit of the work lies mainly in the acute remarks which are freely scattered through it, and in discussions of particular points, which are often of great interest, and for the sake of which alone the work is well worth perusal. As an instance of this we will give a brief account of a single chapter (the twentysixth), which concludes the fourth book on “ Inductive Investiga tion;" it is headed, “Character of the Experimentalist.” After insisting on the impossibility that the efforts of many ordinary men should supply the place of the genius of exceptional men, and remarking that "nothing is less amenable than genius to scientific analysis and explanation," our author goes on to specify some of the mental characteristics of the natural philosopher. His mind must be readily affected by the slightest exceptional phenomena ; his associating and identifying powers must be great ; his imagination active; his powers of deductive reasoning sure and vigorous ; and he must have so strong a love of certainty as to lead him to compare with diligence and candour his speculations with fact and experiment. It is sometimes thought that the philosopher will be cautious in following up trains of speculation; and the notion derives some countenance from the fact, that only successful trains of thought are commonly reserved for publication. But Mr. Jevons points out, from the examples of Kepler and Faraday, that, to use the words of the latter, “in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realized.” He then considers the method pursued by Newton in the Principia' and the Optics' as a type of the true scientific method “of deductive reasoning and experimental verification.” The chapter ends with a notice of a characteristic of the philosop hic mind, to which we have never before had our attention so pointedly drawn, and which is well illustrated by the example of Faraday, viz. the tenacity with which it will cling to a conception as likely to prove true and important in spite of repeated failures to verify it by experiment—which is of course a totally different thing from its being negatived by experiment. Thus Faraday first attempted to demonstrate a relation between magnetism and light in 1822; and though he frequently renewed the attempt, he was unsuccessful until, partly by accident, he obtained a result in 1845. In this case his tenacity was rewarded with success. Another series of attempts to demonstrate a reciprocal relation between gravity and electricity proved unavailing to the end. This instance very appropriately leads up to the remark, that "frequently the exercise of the judgment ought to end in absolute reservation,” the power to maintain this state being yet another characteristic of the philosophic mind. In concluding our notice we will venture to do no more than to mention a single thought which has occurred to us several times while reading the work before us. It is this, that although given trains of reasoning, whether deductive or inductive, command universal assent, yet as soon as we get into a discussion of what constitutes the cogency of the reasoning, we are landed in the region of doubt and debate. This might be thought a paradox were it not so well known to be true. No one doubts the conclusiveness of the deductive reasoning by which Euclid proves his forty-seventh proposition ; but let the question be started, What is deductive reasoning ? and whence does it derive its conclusiveness ? and we shall find the highest authorities giving different answers. A like remark applies to the far more complicated process by which the universal gravitation of matter is proved. Several of Mr. Jevons's logical doctrines might be taken in illustration of these remarks; we will mention one or two. 1. Mr. Jevons, supported by high authority, regards the formula “ Whatever is, is," as a fundamental law of thought—though Shakspeare, to all appearance, regarded it as mere matter for a joke, and Locke treated its pretensions with scorn, and Mr. Mill thought the treatment just*. 2. Mr. Jevons tells us (vol. i. p. 48) that “in ordinary language the verbs is and are express mere inclusion more often than not. “Men are mortals' means that men form part of the class mortal.” There is, of course, a fundamentally different view of the case, according to which the word is merely predicates of men the * It might be supposed that the words “Whatever is, is,” mean “ Whatever exists, exists;" but this is, apparently, not the case, its meaning being “X is X;” e.g. a circle is a circle, or, as Mr. Jevons put it," a thing at any moment is perfectly identical with itself.” So that the clown in Twelfth Night seems to have understood the maxim when he said, “For as the old hermit of Prague, that never saw pen and ink, very wittily said to a niece of King Gorboduc, That, that is, is;' so I being master parson, am master parson. The reference to Locke is Book IV. c. 7, of the “ Essay concerning Human Understanding ;' that to Mr. Mill is p. 408 of An Examination of Sir William Hamilton's Philosophy.' |