376 On Methods of solving certain simple Electrical Problems. resistance of a conductor and the strength of the current produced in it by a given electromotive force, and let it be required to find the strength of the currents which the same electromotive force would generate in conductors whose resistances are respectively represented by the abscissæ O m, and Om. Through M draw a line M P Q parallel to O X, and through m, and m, draw m, P and m, Q parallel to OY; join O P and O Q, and let p and q be the points in which OP and OQ respectively cut Mm; then mp and mq will represent the strengths of the required currents; and if lines be drawn parallel to O X through p so as to intersect m, P in M1, and through q so as to intersect m, Q in M2, M, and M2 will be points whose coordinates, like those of M, represent corresponding values of resistance and strength of current. The points M, M,, and M2, therefore, lie upon the same rectangular hyperbola. In a similar way we may treat many problems of the same sort as those discussed above by aid of what for distinction may be called Ohm's construction; but as the constructions, arising from the choice of resistance and strength of current as coordinates are usually rather more complex than those previously given, and as I have not come across any cases in which they appear to be decidedly more expressive, I will only give two additional examples. To find the permanent resistance and the electromotive force of a battery from observations of the strengths of two currents corresponding to resistances which differ by a known amount.—Let m M (fig. 14) represent the strength of the current when the resistance of the circuit has an unknown value represented by the (unknown) abscissa Om; and let n N express the strength of the current when the resistance has been increased by a known amount denoted by mn. Through M and N draw straight lines parallel to mn, and let N, and M, be the points where these lines respectively intersect n N (produced) and m M. Draw the straight line N, M, and produce it to intersect nm produced in O; then O m represents the original resistance of the circuit, and the rectangle on the base Om with altitude m M, or the rectangle on the base On with altitude n N, represents the electromotive force. The heat produced in unit of time by a constant current of given strength traversing a conductor of given resistance can be represented by the volume of a right square prism, two of whose dimensions represent the strength of the current, while the third represents the resistance; and in the case of a battery of constant electromotive force, the relation between the resistance of the circuit and the heat produced in unit of time can be expressed generally as follows:-Take three rectangular axes, OX, OY, and OZ (fig. 15); in OX take OK to represent the resistance, and in OY take O M so that the area of the rectangle K M represents the electromotive force, and therefore ŎM the strength of the current. Similarly, take ON in OZ also to represent the strength of the current. Then the heat generated in unit of time is proportional to the contents of the rectangular parallelopiped O P, constructed upon the lines OK, OM, and ON. The locus of P is the intersection of two equal and similar hyperbolic cylinders, whose equations are respectively xy=constant electromotive force, and xz constant electromotive force, and is itself a rectangular hyperbola in the plane of O X and OQ, and having these lines for asymptotes. If O K' represent the internal resistance of a battery, and K' K the external resistance, the heat generated inside the battery is represented by the parallelopiped O P', and that generated in the external conductor by the parallelopiped K'P. = = XLIII. Remarks on Helmholtz's Memoir on the Conservation of Force. By ROBERT MOON, M.A., Honorary Fellow of Queen's College, Cambridge*. THE ور HE following observations are offered on the assumption that Helmholtz's memoir" Ueber die Erhaltung der Kraft (Berlin, bei G. Reimer, 1847), the original of which I have not had an opportunity of examining, is correctly represented by Professor Tyndall's translation contained in Taylor's Scientific Memoirs for 1853. I pass by for the present the introductory matter and the disquisition on the conservation of vis viva contained in the memoir; nor shall I now discuss the grounds upon which the author rests his primary induction that the constancy of "the sum of the tensions and vires vive" (which undoubtedly holds " in all cases of the motion of free material points under the influence of their attractive and repulsive forces, whose intensity depends solely on the distance") represents a general law of nature. I come to the author's "special application" of the principle, or to what would be more correctly designated as his attempts to demonstrate its truth in cases of motion where we do not deal with "material particles under the influence of their attractive and repulsive forces," but with continuous masses the different portions of which act upon each other otherwise than by attraction or repulsion; and of these cases of motion I shall confine myself to one, viz. where "a medium. . . is traversed by a train of waves"-a case in which Dr. Helmholtz evidently considers * Communicated by the Author. that "the principle of the conservation of vis viva" ho lor's Scient. Mem. 1853, p. 127.) Now in order to establish the truth of the conserva as that principle is propounded by Dr. Helmholtz-in in order to show that the constancy of "the sum of t and vires viva " holds in the above or any other instar viously necessary that we should derive by independe values for the vis viva and " the sum of the tensions ". to the kinetic and potential energies of a more modern ture, and that we should then show that the sum of sions so arising is always constant, As Dr. Helmholtz, while attempting to prove tha servation of force holds in the case of a train of waves a medium, confines all his efforts to establishing the of the vis viva, taking not the smallest notice of " the tensions," I am led to conclude that he regards as always vanishing in cases of motion of the kind i It is to be regretted that so important a step should passsd over in silence*. In the mean time I assum author relies on the principle-which I have seen enunciated that the mutual normal actions betwee in contact may be neglected in forming the equation The method of estimating the vis viva adopted by holtz in the case we are considering, I believe to be u neous and misleading. At present, however, I do n dwell upon this point, but shall proceed to show, as I strictly : I. That if it were true that when waves traverse the sum of the tensions disappears from, or is const equation of vis viva, that fact would be fatal to the th conservation of force proposed by Dr. Helmholtz. II. That it is not true that the sum of the tensions vanishes under the circumstances supposed. III. That the principle of vis viva, and therefore th of the conservation of force as propounded by Dr. does not hold in the case of waves traversing a medi I. It is well known that a wave in which the normal to the front may be propagated in a cylindrical with air in a direction parallel to the axis, without any change in its length, or in the mode in which densation and particle-velocity are distributed thro * Clausius makes upon Helmholtz's memoir the following "It is to be regretted that the author of this ingenious essay tered more fully into the details of his subject. From this o portions appear to me to be incorrect."-Taylor's Scientific M 1852, p. 6, note. provided that we neglect the friction of the sides of the tube upon the air within it. This will be the case whatever be the form of disturbance, provided only 1. That the velocities and condensations are small, and follow the law of continuity, 2. That the condensation at any point bears a fixed ratio to the particle-velocity at that point (Encyc. Met. Art. Sound, No. 128). Suppose that we have a wave consisting of condensation only which fulfils these conditions; and suppose for simplicity, though this is by no means essential, that the condensation is distributed symmetrically about the middle point of the wave, and that it has a single maximum, which will, of course, be that of the middle point. The particle-velocity throughout will be in the same direction-that, namely, of transmission. Suppose, further, that we have in a different part of the tube a second condensed wave, equal in length to the former, having also but one maximum, viz. at its middle point, where the condensation is equal to the condensation at the middle point of the first wave, and having at equal distances from the middle point on either side of it the same amount of condensation as the first wave at the same distance from its middle point. I shall also suppose that at equal distances from their respective middle points the particle-velocity in each wave is the same in amount but opposite in direction. It follows that the waves will move in opposite directions. Consider the waves, first, as they advance towards each other; next, as after the meeting they overlap; finally, at the period of complete occultation, when the middle point of the one wave coincides with the middle point of the other. Before meeting, the vis viva of either wave will be constant, and the vis viva of the system of two waves will be double that of either taken singly. When the waves overlap, the condensation at any point of the overlapping portions will be the sum of the condensations of the portions superposed, but the particle-velocity at this point will be the difference of the velocities of the superposed portions taken singly. In this part of the system, therefore, i. e. in each element of the overlapping portion of the waves, vis viva will be lost; so that, as the waves after meeting will gradually more and more overlap, the vis viva of the sytem will continually di minish, till, when the position of complete occultation is arrived at, the velocity at each point, and consequently the vis viva of the system, will have wholly vanished. As the waves emerge from occultation, velocity and conse that "the principle of the conservation of vis viva" holds. (Taylor's Scient. Mem. 1853, p. 127.) Now in order to establish the truth of the conservation of force as that principle is propounded by Dr. Helmholtz-in other words, in order to show that the constancy of "the sum of the tensions and vires vive" holds in the above or any other instance, it is ob viously necessary that we should derive by independent methods values for the vis viva and " the sum of the tensions "-equivalent to the kinetic and potential energies of a more modern nomencla ture, and that we should then show that the sum of the expressions so arising is always constant. As Dr. Helmholtz, while attempting to prove that the con servation of force holds in the case of a train of waves traversing a medium, confines all his efforts to establishing the constancy of the vis viva, taking not the smallest notice of "the summ the tensions," I am led to conclude that he regards the latt as always vanishing in cases of motion of the kind in questionin It is to be regretted that so important a step should have be passsd over in silence*. In the mean time I assume that author relies on the principle-which I have seen and he enunciated-that the mutual normal actions between eleme in contact may be neglected in forming the equation of vis The method of estimating the vis viva adopted by Dr. H holtz in the case we are considering, I believe to be utterly neous and misleading. At present, however, I do not des dwell upon this point, but shall proceed to show, as I hope strictly: I. That if it were true that when waves traverse a m the sum of the tensions disappears from, or is constant equation of vis viva, that fact would be fatal to the theory conservation of force proposed by Dr. Helmholtz. II. That it is not true that the sum of the tensions in vanishes under the circumstances supposed. III. That the principle of vis viva, and therefore the of the conservation of force as propounded by Dr. He does not hold in the case of waves traversing a medium I. It is well known that a wave in which the vib normal to the front may be propagated in a cylindrical with air in a direction parallel to the axis, without m any change in its length, or in the mode in which densation and particle-velocity are distributed thro * Clausius makes upon Helmholtz's memoir the following "It is to be regretted that the author of this ingenious essay tered more fully into the details of his subject. From this portions appear to me to be incorrect."-Taylor's Scientific 1852, p. 6, note. thr thro thro thro out T ing o sions waves ner re sing t The ring half of halves the sta Dra point of Whe the hin the case would But superpo be the s |