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expressed by the vanishing of the first r coordinates .... The relation of the two cases to be compared is expressed by supposing the forces of the remaining types +1,... to be the same, so that AY,+, &c. vanish. Thus for every suffix either vanishes or else AV. Accordingly EVAY is zero, and therefore also, by the law of reciprocity, EVAY. Hence, as above,
showing that the removal of the constraint increases the potential energy by the potential energy of the difference of the deformations.
Corresponding to the above theorems for T and V, there are two more relating to the function F introduced by me in a paper printed in the 'Proceedings of the Mathematical Society for June 1873, expressing the effects of viscosity. We have here to consider systems destitute both of kinetic and potential energy, of which probably the best example is a combination of electrical conductors, conveying currents, whose inductive effects, dependent on inertia, may be neglected. The equations giving the magnitudes of the steady currents are of the form
where F is a quadratic function (in this case with constant coefficients) of the velocities, &c., representing half the dissipation of energy in the unit of time, and Y, &c. are the electromotive forces. It is scarcely necessary to go through the proofs, as they are precisely similar to those already given with the substitution of F for T, and steady forces for impulses. The analogue of Bertrand's theorem tells us that, if given electromotive forces act, the development of heat in unit time is diminished by the introduction of any constraint, as, for example, breaking one of the contacts. And by comparison with Thomson's theorem for initial motions we learn that, if given currents be maintained in the system by forces of corresponding types, the whole development of heat is the least possible under the circumstances (Maxwell's Electricity and Magnetism,' § 284). And precisely as before, we might deduce corollaries relating to the effect of altering the resistance of any part of the combination.
XXVI. Notices respecting New Books.
Exposition Géométrique des Propriétés Générales des Courbes. Par CHARLES RUCHONNET (de Lausanne). Troisième édition, augmentée et en partie refondue. Paris: Gauthier-Villars. Lausanne: Georges Bridel. Zurich: Orell, Fuessli et Comp. 1874 (8vo, pp. 160).
HIS work contains a very complete account of the properties of curved lines which may be considered general, as distinguished from those possessed by certain curves at individual or singular points. Though the author has occasion to notice that there may be points of inflection, he does not consider them, as inflection cannot be a property enjoyed by the points of a curve generally. Thus the proof of art. 44, that in general a curve cuts its osculating plane at the point of contact, consists in showing that if the curve do not cut the osculating plane at an assigned point, there must be inflection at the corresponding point of another
The consequence of this is, that the book consists mainly of a discussion of curvature, and of the properties of the osculating circle and of allied subjects. Thus the five sections which make up the first part of the work (that devoted to plane curves) treat of the tangent, curvature, the osculating circle, and expressions for different magnitudes: viz. (1) those which arise out of a consideration of an infinitely small arc and the tangents at its extremities, such as that the difference between the arc and its chord ultimately equals one 24th of the square of the angle of contingence multiplied by the length of the arc; (2) those which depend on the difference between the radii of curvature at two infinitely near points of the curve, such as that the difference between the angles contained by the chord and the tangents at its extremities is ultimately equal to one half of the square of the angle of conαρ
tingence multiplied by
In the second part (that devoted to tortuous curves, and which occupies about three quarters of the volume) the subjects are much the same; but, of course, many more points come under notice; and, besides, there is an account of Ruled and Developable Surfaces, sufficient to enable the reader to understand the properties of the polar and rectifying surfaces of these curved lines.
Our author's treatment of the subject as a whole is both minute and exact; and though we do not profess to be acquainted with the whole literature of the subject, it is only fair to say that we do not know any work which contains an account of it to be compared with that before us in completeness. The chief peculiarity, however, of the treatment is in its method, which is wholly that of limits each proposition being separately proved by reasoning directly from a diagram, without the use of any of the elementary formulæ of the differential calculus. Thus nothing so recondite Phil. Mag. S. 4. Vol. 49. No. 324. Maech 1875.
as Taylor's Theorem is employed from one end of the book to the other, though the author comes rather near to using it in art. 46. It is unnecessary to add that such a treatment of this subject requires considerable powers of geometrical exposition; and these are certainly possessed by the author. In fact he gives us to understand that he has worked out the whole subject from his own point of view with not much more aid from the labours of his predecessors than what is implied in the fact of their having established by a different method all the leading facts of the subject.
The advantage gained by a purely geometrical exposition of such a subject as Curvature is, that the student learns from it what in actual space corresponds to his algebraical formulæ ; and this is a matter to which his attention has often to be directed. On the other hand, it is liable to the somewhat serious drawback, that the results are obtained by the use of an instrument of research inadequate to the purpose, unless in a very skilful hand, while they can easily be got at by other means. Suppose a student to have a moderate skill in analysis, and to know the few general formulæ relating to the subject which are to be found in all books of solid geometry; e. g., suppose him to have mastered pp. 407417 of De Morgan's Differential Calculus,' it is hardly too much to affirm that he would find it easier to investigate by their means most of the theorems contained in M. Ruchonnet's book than to make out his proofs. He would observe that such magnitudes as the angles of contingence and torsion, the radius of curvature, the ratio of its increment to that of the arc- dp in fact-are intrinsic
to the curve, and the relations between them independent of the coordinate axes chosen. Consequently he would choose the axes so as to simplify, as much as possible, the general expressions given in books; and, as a rule, the required results would then be obtained without much difficulty. Take for instance the proposition quoted above, that in general the curve cuts the osculating plane at the point of contact. Suppose the curve to be given by the equations y=F(x) and z=f(x), that the point under consideration is taken as the origin, the osculating plane as the plane of xy, and (though this is not necessary for the purpose immediately in hand) the tangent as the axis of a. It follows from these suppositions that F(0), F'(0), ƒ(0), ƒ'(0), and ƒ"(0) are severally zero. Now consider a point (h, k, l) near the origin; we have
k=F(h)=F"(0). h2+†F'''(0). h3 +
The second equation shows that in general changes its sign with h, i. e. on one side of the normal plane the curve is above, and on the other side below the osculating plane, which it therefore cuts at the point of contact. Our author's proof of this theorem is most ingenious, but, as already mentioned, is indirect, and by no means easy to follow.
As a further illustration of our meaning we will consider a property of plane curves, which is not noticed in the usual text-books, and which M. Ruchonnet attributes to M. Abel Transon (p. 42); viz. that if 0 is the angle between the normal and the diametral curve at the same point of a given curve,
Let y=f(a) be the equation to the curve, the point under consideration (O) being taken as the origin, and the axis of a so chosen as to touch the curve; consequently f(0) and f'(0) are severally zero, Suppose the curve to be cut by a chord PP' parallel to Ox; if A is its middle point, the ultimate value of AOy is 0. If (h, k) and (-h', k) are the coordinates of P and P', it is plain that tan ✪ is the ultimate value of (h-h')÷2k. Now, observing that it is not necessary to retain terms above the third order, we have
Therefore, subtracting and dividing out h+h', the ultimate value
But it follows, from the well-known general
expression for the radius of curvature, that
at the origin; and this proves the theorem.
Éléments de Calcul Approximatif. Par CHARLES RUCHONNET (de Lausanne). Seconde édition augmentée. Paris: Gauthier-Villars. Lausanne: Georges Bridel. Zurich: Orell, Fuessli et Comp. 1874 (8vo, pp. 65).
The question discussed in the work before us is this:-What precautions must be taken in a numerical operation to ensure that the first n digits of the final result shall be exact? The author considers separately the operations of addition (and subtraction), multiplication, division, extraction of roots, and, very briefly, the case of a function of one variable. He illustrates his rule by working out several examples; but he does not insert examples for practice, as an English writer would probably have done. The above-mentioned operations become complicated when any or all the numbers concerned are incommensurable; and in these cases a second question arises, viz. to what degree of approximation these numbers must be taken separately to ensure the required degree of accuracy in the final result.
We shall perhaps convey the best idea of the book by descri
bing briefly our author's treatment of the question of multiplication, to which his third chapter is devoted. In the first place he explains, by means of two examples, Oughtred's method of contracted multiplication; but instead of leaving the last figure uncertain as is usually done, he notices that the process gives an approximation in defect, and points out that if the sum of the digits in the multiplier, which give partial products, be increased by the first unused digit of the multiplier, and by unity if there be a second unused digit in the multiplier, and then this sum be added to the product, we now have an approximation in excess; and by comparing the two we obtain a result in which a certain number of digits are known to be exact. Moreover the rule, as usually stated, directs that if, for instance, the result were required to be true for two places of decimals, the unit digit of the multiplier should be placed under the second decimal digit of the multiplicand; our author notices that it should usually be placed under the third decimal digit, and in certain circumstances under the fourth or fifth, and so on in other cases. He next enters on the question, Given that the factors are incommensurable numbers, to what degree of approximation must they be known that the first n digits of the product may be exact? He first shows that when the factors are approximate by defect, the relative error of their product is less than the sum of the relative errors of the factors; and then reasons as follows:-Suppose that there are p factors (p being less than 10), calculate each factor to n+1 digits; then the relative error of each factor is less than and consequently the sum P.10 of their relative errors will be less than and their product will 10"
have the first n digits exact. He also observes that if the first digit of the required product is known before hand, it is, under certain circumstances (which he specifies), enough to calculate some of the factors to n digits.
It will be evident from this that the author is quite justified in thinking that "he has given completeness, in the present work, to methods laid down by other writers." He states that the work was originally published as an appendix to that on curved lines noticed above, that it has been carefully revised, and contains several important additions; amongst others is a complete solution of the question, How many digits of a number must be known in order that its mth root may have its first n digits exact?