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 Om, and Om. Through M draw a line M P Q parallel to O X, and through m, and m, draw m, P and moQ parallel to OY; join OP and O Q, and let p and 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 OX through p so as to intersect m, P in M,, and through q so as to intersect m, Q in M,, M, and M, 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. q 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 m n, 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 Om 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 OM 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 x y = constant electromotive force, and xz=constant=electromotive force, and is itself a rectangular hyperbola in the plane of O X and O Q, 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 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" 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 viva" holds in the above or any other instance, it is obviously 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 nomenclature, 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 conservation 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 sum of the tensions," I am led to conclude that he regards the latter as always vanishing in cases of motion of the kind in question. It is to be regretted that so important a step should have been passsd over in silence*. In the mean time I assume that the author relies on the principle-which I have seen and heard enunciated that the mutual normal actions between elements in contact may be neglected in forming the equation of vis viva, The method of estimating the vis viva adopted by Dr. Helmholtz in the case we are considering, I believe to be utterly erroneous and misleading. At present, however, I do not desire to dwell upon this point, but shall proceed to show, as I hope to do strictly : I. That if it were true that when waves traverse a medium the sum of the tensions disappears from, or is constant in, the equation of vis viva, that fact would be fatal to the theory of the conservation of force proposed by Dr. Helmholtz. II. That it is not true that the sum of the tensions in general vanishes under the circumstances supposed. III. That the principle of vis viva, and therefore the principle of the conservation of force as propounded by Dr. Helmholtz, does not hold in the case of waves traversing a medium. I. It is well known that a wave in which the vibration is normal to the front may be propagated in a cylindrical tube filled with air in a direction parallel to the axis, without undergoing any change in its length, or in the mode in which the condensation and particle-velocity are distributed throughout it, * Clausius makes upon Helmholtz's memoir the following remark :-"It is to be regretted that the author of this ingenious essay has not entered more fully into the details of his subject. From this cause certain portions appear to me to be incorrect."-Taylor's Scientific Memoirs, Nov. 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 diminish, 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 quently vis viva will constantly be generated, till, as the waves become extricated the one from the other, the vis viva will be, what it will ever after remain, exactly what it was at first. Hence, if it were true, as Dr. Helmholtz assumes, that the normal actions disappear from the equation of vis viva when we consider the internal motions of continuous masses, that fact alone would suffice to demonstrate the entire failure of his principle of the conservation of force in cases of motion such as that we have been considering. II. That the sum of the tensions does not in general vanish when a wave traverses a medium may be seen as follows: Let x and y be the ordinates (measured parallel to the axis of the tube) in the state of rest and at the time t of one surface of an element made by planes perpendicular to the axis, p the pressure at the same surface, D the density of equilibrium. The equation of motion may be put under the form dp whence, multiplying by dt and integrating, we get for the equa dt the integration being effected between any limits we may fix upon. Suppose that we have two waves of condensation, such as those already described, travelling in opposite directions; and (x and y being measured parallel to c AC in the annexed figure) let the ordinate perpendicular to c A C of the curve A B C represent the condensation in the wave which moves towards the right, in which direction we will suppose x and y measured positively; while the corresponding ordinate of the curve a b c represents the condensation in the wave which moves to the left. Assuming the truth of Boyle's law in cases of motion (which it will suffice to do for the purposes of this paper), if the wave corresponding to ABC (or, as for brevity I shall designate it dp the wave ABC) stood alone, we should have positive through |