Hence for the ratio of the radii vectores from the two poles for the points P, Q, R, S, we have Similarly, we should find for the points P1, Q1, and R, the common ratio ·m+1 and for the points Q', R', and S' the common ratio Hence, not only are the equipotential lines for two equal and opposite poles characterized 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. Let r be the distance of the given point from the source, and 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 and since the source and sink are equal, V'1 = -V, and QQ; therefore the resultant potential, or V+V', is 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 log, which agrees with what was proved above (§ 18). We may write (4) thus (§ 8), Q U= log μ2=n. Av; 2πκδ whence, regarding the source and sink for the present as mere points, the value no will correspond to the former, and n=∞ to the latter, while n=0 denotes the straight line at equal distances from both. 20. Let the distances from O (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 I and l'; then, a being as usual half the distance between the poles, a+l l'+a where μ is the ratio of the radii vectores of the given circle. Hence For the radius of the circle, we have evidently also p2=CA.CB=(d+a)h=d2 — a2, 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 BC and either BD or B D'. Now BC=h=d-a= 2a Hence for the ratio of the radii vectores from the two poles for the points P, Q, R, S, we have Similarly, we should find for the points P, Q, and R1 the common ratio -m+1 and for the points Q', R', and S' the common ratio Hence, not only are the equipotential lines for two equal and opposite poles characterized 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 (=μ) 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. Let r be the distance of the given point from the source, and 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 and since the source and sink are equal, V1 = 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 log, which agrees with what was proved above (§ 18). We may write (4) thus (§ 8), Q U= log μ" =n. Av ; whence, regarding the source and sink for the present as mere points, the value no will correspond to the former, and n=∞ to the latter, while n=0 denotes the straight line at equal distances from both. 20. Let the distances from O (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 7 and l'; then, a being as usual half the distance between the poles, a+l l'+a = =μ", a- τι -a where μ is the ratio of the radii vectores of the given circle. Hence For the radius of the circle, we have evidently p2=CA.CB=(d+a)h=d2 — a2, 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 Any convenient value such as 2 or may be given to μ; or if a 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 l, and m the number of lines required on each side of O. We shall have a+L which determines the value of μ. 1 a+L m The successive values of B D' are then a-L' The successive values of BC are simply alternate values of these, namely Plate IX. shows the equipotential lines drawn for a value of μ=33. [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. 8vo, pp. 463 and 480). THIS 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 Igreat 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, |