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added to the circuit in order to get a second reading of the galvanometer; then the pulley P' is adjusted so that the cord passing over it may indicate upon the quadrant B the galvanometer-deflection obtained after introducing the resistance cor. responding to A A'; and lastly the vertical scale is moved so that a vertical line drawn through the centre of the divisionmarks and continued downwards to meet the horizontal scale at O may be exactly behind the point Q, where the cords A Pand A'P' cross each other. We have then only to read off the horizontal and vertical distances, A O and O Q, to get the permanent resistance and electromotive force of the circuit*. To adapt the apparatus to use with a sine-galvanometer, it would be only necessary to replace the vertical scale 0 Q by a quadrant marked with concentric circular arcs, each differing from the next by 1 millim. in radius.
The following additional examples may be given of the same mode of treatment in connexion with allied problems :
Development of Heat in a Galvanic Circuit.-Draw OC (fig. 6) horizontally to represent the resistance of the circuit, and OA vertically to represent the electromotive force; from A draw AC and also AD at right angles to AC; then OD represents the beat produced in the circuit per unit of time. It is of course to be understood that O A denotes not necessarily the electromotive force of the battery, but the resultant electromotive force of the whole circuit—that is, the algebraic sum of all the electromotive forces which act anywhere in it. If OB denote the resistance of the battery and BC the remaining resistance of the circuit, and if B A' be drawn vertically, A'EF horizontally, and A'D' at right angles to A C, E F will denote the heat generated in the battery, and B D the heat generated in the external part of the circuit.
Strength of Currents in the different branches of a divided Circuit. - In considering this problem it is needful first of all to have a mode of representing geometrically the combined resistance of two or more conductors connected in multiple arc." Let A B and B C in the same straight line (fig. 7) represent the resistances of two conductors taken separately. Draw A B' equal to A B and at right angles to it; join B'C, and through B draw a line parallel to A B' and cutting B'C in L; then B L represents the combined resistance of the two conductors. The same process is applicable to any number of conductors. Thus, let AB, BC, and CD (fig. 8) represent the separate resistances of three conductors connected in multiple arc. Making the con
* The quadrant A is made so that it can be turned aside to allow of the vertical scale being brought close up to the zero-point of the horizontal scale when low battery-resistances bave to be measured.
struction indicated above, BL represents the resistance equivalent to A B and B C in multiple arc. Draw L M' parallel to ABCD, make CM=B L, draw M M' at right angles to B C, join MD, and draw C N perpendicular to CD, CN then represents the joint resistance of the three conductors.
Now consider a conducting-system such as that indicated in fig. 9, where two points, P and Q, are connected through three conductors whose resistances are respectively r, r1, and rą; and let a battery of electromotive force e make part of the first conductor. The strength of the currents and the distribution of potential in the various parts of the system can be represented as follows :- Take 0 A (fig. 10) to represent r, from O draw O B and 0 C in opposite directions perpendicular to 0 A, make CC' equal to 0 C and perpendicular to it, draw B C cutting A 0 produced in D; then O D represents the joint resistance of my and rą. Draw A E perpendicular to A O to represent the electromotive force e, and join E D. Then tan LADE represents the strength of the current in the battery and the sum of the currents in the two parallel branches. These may be obtained separately thus : let the point of intersection of ED and OB be denoted by F; then 0 F represents the difference of potential between the points Pand Q or the electromotive force which is effective in the two conductors of resistance r, and rg. In 0 A make OF=O F and draw F B and FC; we have then, for the strength of the currents in the conductors whose resistances are represented by 0 B and 0 C, tan ZOBF' and tan 20CF respectively.
Next let two of the conductors connected together at P and Q contain galvanic batteries, and, as before, let the resistance of the branch containing one battery be r, while that of the branch containing the other is rı, and let the electromotive forces be e and e, respectively. If the batteries are so connected that both tend to make the potential at P differ from that at Q in the same sense, we have an arrangement of which a special case is presented by Poggendorff's “con pensation method” for the comparison of electromotive forces. To obtain a geometrical expression for the strengths of the currents in the various parts of the circuit in the general case (that is, without assuming that there is "compensation" in any branch), we may proceed in the following manner :- Take 0 Å and 0 B in the same straight line (fig. 11) to represent the resistances r and r, of the two branches including the batteries, and 0 C at right angles to A B to represent the resistance r, of the third branch. In CO produced inake 0 A,=0 A, and o B,=0 B, also draw A, A, equal and parallel to 0 A and B, B, equal and parallel to 0 B, and join A, C and B,C. Let A, C cut o A in N, and let B,C
cut O B in M. From A draw A E at right angles to A O to represent the electromotive force e, and draw E M cutting 0 C in E'; similarly, from B draw B F* to represent the electromotive force e, and draw F N cutting CO (produced) in F'. Then F' E'=FO+OF represents the electromotive force which is effective in the conductor of resistance rą represented by 0 C, and AE-FE and F B-FE represent the electromotive forces which are effective in the branches of resistance r and ri respectively. III. Ordinates represent Strength of Current, and Abscisse
represent Resistances. With this system of coordinates, electromotive force is expressed by the area of a rectangle. Thus, if a given battery produces a current whose strength is represented by the ordinate Mm of the point M (fig. 12), in a circuit the resistance of which is represented by the abscissa 0 m of the same point, its electromotive force must be proportional to the area of the rectangle OM; and if the battery is “constant,” the currents, represented by the ordinates M, m,, M, me, and corresponding to the resistances denoted by the abscissæ 0 m, 0 mq, will be such that the areas of the rectangles OM, OM,, and O M, are all equal; and hence the characteristic property of the battery will be ex. pressed by the curve which is the locus of the points M, M, &c.—that is to say, by a rectangular hyperbola whose asymptotes are the axes of no current and no resistance. When the hyperbola characteristic of a given battery is drawn, it is of course easy by measuring coordinates to find what current would flow through a given resistance, or conversely to find what must be the resistance of the circuit in order that the current may have a given strength; but the difficulty of tracing an hyperbola with accuracy greatly lessens the practical utility of this method of calculation. Since, however, the asymptotes are fixed, each hyperbola is completely defined when one point of it is given ; and, in like manner, when the corresponding values of current and resistance are known in any one case for a given battery, each is determined for any other case when the value of the other is given. Accordingly the actual drawing of an hyperbola is not necessary; for when one point is assigned, any other points corresponding to given problems can be easily found.
For instance, let the coordinates of the point M (fig. 13), referred to the axes 0 X and O Y, represent respectively the 876 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 0 m, and 0.mą. Through M draw a line M P Q parallel to 0 X, and through m, and m, draw m, P and mąQ parallel to OY; join O P and Q, and let p and q be the points in which O P and O Q respectively cut Mm; then mp and my will represent the strengths of the required currents ; and if lines be drawn parallel to 0 X through p so as to intersect m, P in M,, and through q so as to intersect me 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
* BF must drawn in the opposite direction A E if, as supposed above, the batteries are so connected that the difference of potential between the poiuts P and Q due to each battery separately is of the same sigt
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 constructious, 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 0 m; 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 0; 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, O X, O Y,
and O Z (fig. 15); in 0 X take OK to represent the resistance, and in O Y take OM so that the area of the rectangle KM represents the electromotive force, and therefore OM the strength of the current. Similarly, take O N in O Z 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 OP, 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
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.