within 1 or 2 lines of the sphere, the streaming out is very violent, and the remaining density of the sphere only sufficient for sparks of the same length. If the point be withdrawn to a greater distance from the sphere, the streaming out diminishes, but not to an extent sufficient to leave the sphere a density necessary for sparks of more than 2 lines in length. The sparks do not appear, and the same is true of all larger distances. If the point be withdrawn so far that all streaming out ceases, the distance must be much greater than 4 inches, and then the unenfeebled density of the sphere is insufficient for sparks of such a length. This process also takes place when the form of the body brought near the sphere is different sparks are obtained up to a certain length, and none beyond this. Only by a definite relation of the action of the machine to the magnitude of an approximated sphere can the pause phænomenon occur between two spheres, as observed in the experiment of Nairne. Such an experiment is not to be repeated with certainty. But the effect is certain, because independent to a greater extent of the action of the machine, when the truncated cone (the pause-cone) is made use of. At the closest approximation of the cone to the sphere, the augmentation of the density, and the consequent streaming out of the electricity, is not sufficient to hinder the production of sparks. On the diminution of this density with increasing distance, the sparks could never disappear if the disappearance were not effected by the increased number of the emissive points, or in other words, by the breadth of the luminous segment of the sphere. At the shortest distance at which the sparks are absent (9 lines) this breadth is not considerable, but it augments with the increasing distance until the density throughout the extent of the segment has sunk below the limit necessary for the streaming out of the electricity. From this point forward the width of the segment diminishes. At the distance of 11⁄2 inch, where the luminous segment is about an inch wide, the density upon it had sunk so far that the tranquil glow was no longer seen alone, but from the edge of the segment single luminous fibres issued in the form of a brush. At a greater distance, therefore, the sparks appeared again, and continued so long as the conductor yielded sparks. If the form of the cone be altered only a little by rendering the frustum shorter, sparks are obtained at all the distances mentioned; if it be made a little sharper, sparks are obtained up to a certain point, and none afterwards. In like manner no pauses were obtained with larger and smaller spheres than those for which the cone has been arranged. We convince ourselves easily of the change in the arrangement of the electricity on the cone, and the extinction of the pauses thereby produced, if during the experiment a conducting body is brought near the sphere, while a sphere is placed near the prime conductor at a suitable place. Under the large sphere, A, of the conductor (fig. 1) a small sphere was placed at a distance of 9 lines, and the pause-cone removed to a distance of 1 inch from the pause-sphere. By the most persistent turning of the machine no spark appeared at either of the intervals. When, however, the hand was approached from above to within a distance of 10 inches, sparks passed incessantly to the lower sphere A; and, when the hand was removed, again disappeared. In the dark it was observed that by the approximation of the hand the luminous segment was extinguished upon the pause-sphere. Here, therefore, by the approximation of a conductor, the arrangement on the sphere was changed, by this the streaming out of the electricity prevented, and the quantity of electricity remaining on the conductor augmented. The electricity of the pausesphere is not only diminished by the streaming out of the electricity, but also by the circumstance that the brush of the adjacent pause-cone sends air negatively electrified against it. This circumstance cannot, however, cause the absence of the sparks, which was at once shown when the conductor of the machine was negatively electrified. Although the machine yielded negative electricity in a less quantity than positive, and although here the pause-cone sent a much longer brush against the sphere, still sparks varying from of an inch to 2 inches in length were easily obtained; which were absent when the conductor was charged positively. The phænomenon of the pauses was not in this case perfectly exhibited, being characterized by the fact that at some distances between the electrodes (1, 14, 14 inch) the sparks passed tardily and irregularly, and were sometimes mixed with hissing brushes; while, when a conductor was brought near, the regularity of the process was established. As there is no doubt that the arrangement of the negative electricity on the sphere was precisely the same as that of the positive in the former experiments, these irregularities in the pauses would be very surprising, were we not aware of the fact, that in free air negative electricity is brought with much more difficulty to the state of glow than positive. Thus at the rounded end of a metallic cylinder, 0-3 of an inch in thickness, Faraday easily obtained the glow with positive electricity, but failed to obtain it with negative electricity. (Experimental Researches, 530.) The imperfection of the pauses with negative electricity, is due to the difficulty of obtaining the glow with this kind of electricity, and furnishes an additional proof that the pauses are determined by the glowing of the sphere. In the explanation of the pauses just given, one difficulty remains untouched, which appears to be of general interest, inas much as it shows itself in other experiments which are frequently made. The absence of the sparks in the interval of the pause has been ascribed to the circumstance, that the portion of the spherical surface nearest to the cone, possesses a greater density than that necessary for the production of a spark. Now the electricity developed on the conductor has not attained this density instantaneously, but gradually, and there must have been a point in which the sphere possessed the exact density necessary for a spark. The question is, why did not the spark make its appearance at this particular time? The same question may be asked with reference to experiments which are better known. A large sphere on the conductor of a machine gives sparks to an approximated conductor; a small sphere gives a brush. When through the slow turning of the machine a brush is produced at a particular point of the conductor, this may frequently be transformed by quicker turning into a glow. A smooth sphere which furnishes sparks, gives brushes when it is roughened, and so forth. In all these cases the same question suggests itself, why a phanomenon of higher density appears, and that of lower density disappears, while that of intermediate density does not occur, although the density has been gradually increased. This difficulty M. Riess believes renders the assumption necessary, that, for the production of a spark, not only is a definite density requisite, but that this density must exist for a certain time. According to this, the spark first breaks out shortly after the electrodes have attained the density necessary to its production. This time may be small, but it is at all events very great in comparison with the time required by the electricity to arrange itself upon a good conductor; and thus it is not surprising that at a place which has attained to a higher density, instead of the spark another mode of discharge should make its appearance. In the same manner, time is necessary for the brush and the glow to be produced; and it must be stated generally, that an obstacle is to be overcome before the electricity upon the surface of a conductor can propagate itself to the nearest layer of the limiting medium, and that this overcoming requires time. This assumption agrees well with the fact of the resistance to the passage of an electric current from one medium into another. In the experiments on the pauses, the pause-sphere receives, at all distances of the cone, more electricity than is necessary to produce the density required by a spark; but a portion of this electricity is lost by streaming out. At smaller distances of the electrodes the portion streaming out is of great density, but only of limited extent; and the density necessary for a spark is but small; hence the density of the sphere does not sink lower than what is required for the appearance of the spark. At the great est distance of the pause-cone (24 inches), the place streaming out is small and of inconsiderable density; here, therefore, the sphere also retains the density necessary for a spark. When, on the contrary, the sphere stands within the pause-distance, the extent of the surface streaming out, and its density are both great, and the density of the sphere sinks immediately from its highest value till it falls below that which is requisite for a spark. It may be seen from this, that the sparks where pauses exhibit themselves must be preceded by a streaming out, which terminates in a sounding brush. In some positions of the cone this streaming out may be detected by the eye, in others it is only made manifest to the ear by a peculiar noise which accompanies the sound of the sparks. The visible brushes which alternate with sparks, appear near the beginning and the end of the pauses, where the narrowest and longest sparks are observed. It occurs here sometimes, that, at the commencement of the experiment sparks are obtained, but during the continuance of it, none. To obtain sparks again, in such a case, the experiment must be discontinued for a few minutes, or still better, the conductor must be electrified negatively for a few seconds, and then the experiment continued. The absence of the sparks is here due to the electrifying of the air, which alters the electric arrangement on the pause-sphere, and which is neutralized when the air, by the accession of the opposite electricity, has become again non-electric. XL. On the Problem of the In-and-circumscribed Triangle. By the Rev. GEORGE SALMON, Trinity College, Dublin*. THE following is a direct investigation of the solution which I gave in the last Number of this Magazine of the problem, "to find the locus of the vertex of a triangle two of whose angles move on a conic V, and whose three sides touch a conic U." I first form the equation of the pair of lines drawn from any point to touch U; then the equation of either pair of lines joining the points where these tangents meet V; and lastly, form the condition that one of those joining lines should touch U. I write the condition that AU+V shall represent right lines, and if V represents right lines, A'=0; and if, moreover, one of these right lines touch U, we must have 2=4AO'. Now if P represent the polar of any point, the equation of the pair of tangents drawn from that point to U is UU'—P2=0. To form the equation of the pair of lines joining the points where * Communicated by the Author. these tangents meet V, we must determine λ so that UU'-P2+λV may represent right lines, and we get A'x2+Fλ+AU'V'=0, where F has the meaning given in the last Number, and 'Conics,' p. 268. The condition that one of these lines should touch U is (2AU'+x®)2=4A(AU22+λ(®U'+AV')+®'λ2), which reduces to 4AoV' +λ(4A®'—®2)=0. Eliminating λ, the result becomes divisible by V, and is 16A3A'V+4A(℗2—4A®′)F+ (O2—4A®')2U=0, as was found before. It is obvious that the same method would readily give the equation of the locus, if the third side instead of touching U touch a new conic W. If the three sides touch U, one vertex moves on V and another on a right line L, then the equation of the locus of the third vertex is where (A6-MU)2=AU¥2, is the result of elimination between the equations L, V, dU dU' d U' and x +y. +2 ;M is the condition that the pole of the dx dy dz line L with respect to U should lie on V; and is the equation of the polar with respect to V of the pole of L with respect to U. I have formed the equation of the locus when two vertices move on conics, and the three sides touch the conic 2+2xy; but I do not perceive the rule for forming the equation in the general case, except that I can verify (what Mr. Cayley has proved geometrically) that the equation is of the form o2+U=0, where d U' dU' dU' is obtained by eliminating between x +y +2- and the equations of the conics traversed by the two base angles. If the three vertices move on V, the three sides may touch respectively the conics U+aV, U+6V, U+cV, provided that a, b, c are connected by the relation dx dy dz {℗−(ab+ac+bc)▲'}2 = {2®'+2(a+b+c)▲'} {2A+2abc▲'}. The problem to find the locus of the vertex of a quadrilateral, whose other three vertices move on V, and whose sides touch U, reduces itself to a problem noticed already. I use the following abbreviations :— 4AA'=a; 2-4A0'-B; 2Aα+B=y. |