VI. Calculation of the Absolute Resistance. If we insert these numbers for A, B, λ, T, and further those for λo, to, K, and S in the formula (page 305) we obtain for the absolute resistance of the inductor and galvanometer for the four determinations, taking as unit the resistearth-quadrant ance 1010 millim. second' or briefly second Ia. w= earth-quadrant The greatest difference of these numbers corresponds to a change of temperature in the wire of less than 2°, which cannot be controlled. VII. Comparison of the Resistance w with Siemens's Standards. The two standards, Nos. 1135 and 1143, each of which contained 4 mercury units, were guaranteed to be correct at 19°4 and 18°3 respectively. The increase of the resistance in the German-silver wire amounted for 1° to 0-0004 of the whole. Hence the ratio, if both were at the same temperature, must have been No. 1143 No. 1135 =1·00044. I found, by comparison with third resistances, 1.00050, 1.00046, 1.00055; mean=1.00050. The difference from the above ratio corresponds to an error of temperature of only 0° 15, and may therefore serve as a test for the accuracy of the copies and of the method of comparison. At the beginning of the observations I did not possess the standards; and instead of them I constructed four provisional German-silver resistances (I will denote them as A, B, C, D), each nearly equal to 4 Siemens's units, and which in the determination I. served for comparison with the inductor. They were subsequently compared with the standards, which had meanwhile reached me. A=4·1021 B=40977 C=4·1095 D=40965 Siemens. For interpolation two tenths of a Siemens unit were necessary. They were constructed of two pieces of German-silver wire, soldered in lengths of 325 millims. to copper rods. As it was found, moreover, that 3250 millims. of the same wire had the resistance 1 Siemens at +12°.0, the pieces in question accurately represent tenths at that temperature. As the circuit inductor + galvanometer had been brought to nearly 4 Siemens by the introduction of a small load of copper wire, a differential galvanometer could be used. It was externally similar to that described in page 345 et seq. To be quite free from thermal influences, the comparison was made by means of short currents produced by Weber's magnet inductor by the method of multiplication, and which were divided by the two resistances and the two galvanometer wires. The heating by the very feeble brief currents is in any case vanishingly small; at the same time accidental thermo-electromotive forces, which in a circuit extending over a large space are unavoidable, have no influence, as the induction-currents alternate in direction. The method used, especially with reference to the extra current produced in the inductor, has been described by me in Poggendorff's Annalen, vol. cxlii. p. 418; and I refer to that paper. The arrangement of the whole of the instruments is represented schematically, fig. 2. C is a stopper commutator with six copper plates made of ebonite, massive, and very carefully worked. Near it are five solid clamping-screws, which can be placed in conducting communication with each other by a slider. The connexion of the other parts is shown by the figure. J and G, terrestrial inductor and galvanometer, form the circuit the absolute resistance of which is to be determined. These are closed at 1 by stoppers. For comparison with Siemens's standard, 1 is removed and 2 stopped. The standard E is then connected with one, and the circuit J G with the other branch of the differential multiplier I). To change them in reference to these branches, it is simply necessary to take out the stopper at 2 and insert it at 3. M is the source of the current, the magnet inductor (which was removed during the absolute measurements). The tenths added to E or JG are denoted by 0.1; by sliding and fixing the shunt they become ineffective. The needles of the galvanometer G were, of course, fixed during the comparison. It was specially ascertained that the motion of the inducing magnet in M, which, as is well known, consists of two magnets with similar poles presented to each other, had no action at a distance on the inductor. It is useless to detail the experiments, since the errors of comparison cannot in any case compare with those of absolute determination. Moreover the example deduced in the memoir already cited (p. 421) is one of these determinations. Nos. Ia. and Ib. of the absolute measurements belong to a comparison made between them with the provisional standards, as the Siemens's standards had not then reached me. II. and III. were made on other days, when a comparison was made both before and after. The experiments gave the resistance w, or the inductor + galvanometer, equal to the following numbers in Siemens's units: I. Temperature of the standard = +15°·3. A+0.00714-1034 Siemens. Comparing these resistances expressed in Siemens's units with those expressed in absolute measure, we obtain I. 4-1029 Siemens =3.9812 carth-quad. ; 1 Siem. 0.9703 earth-quadrant *. In the mean, therefore, 1 Siemens's mercury unit =0.9717 With regard to the ratio of the British-Association unit to Siemens's unit, the most trustworthy value hitherto published is probably that which M. Dehms deduced from a comparison by Mr. Fleeming Jenkin†, 1 British-Association unit=1.0493 Siemens. M. Dehms and Mr. Hermann Siemens had the goodness to make at my request a new comparison, from which a BritishAssociation unit (No. 61) belonging to Siemens's laboratory was * In the result 0-9705 (Gött. Nach. 1870, p. 523) an error of calculation had crept in. + Pogg. Ann. vol. cxxxvi. p. 404. Report of the British Association, 1864, p. 349. Phil. Mag. vol. xxix. p. 477. Phil. Mag. S. 4. Vol. 47. No. 313. May 1874. 2 A found to be 1.0473. Owing to an injury to the unit, the comparison had to be made in the air; and therefore no great weight must be assigned to it. The British-Association units belonging to M. Brix (No. 21) and M. Weber (No. 51) were also compared, and gave, in accordance with the above, the value 1.0493. If this agreement be compared with the former enormous differences in the statements regarding units of resistance, a very satisfactory proof is afforded as to the progress in this kind of measurement*. Taking the number 1.0493, it will be found that this unit is nearly 2 per cent. more than was intended. The electromotive forces of Daniell and Grove I have found, in conjunction with Ammann, to be 1171 and 19.98 SiemensWeber units respectively; they have therefore the absolute value+ Daniell = 11.38 × 1010 millim. milligrm. Grove = 19.42 x 1010 second2 The thermoelectric force of German-silver-iron, expressed in the same units, 2400000 for a difference in the junctions of one degree at mean temperatures. XLIII. Direct Solution of a Geometrical Problem. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN, MAX Gonville and Caius College, Cambridge, March 27, 1874. JAY I request the publication in your Magazine of the accompanying paper and letter which came from a lady in California? The paper, as you will see, is a solution of a geometrical problem which, more than twenty years ago, excited considerable interest by its discussion in your Magazine. The Vice-Chancellor of Cambridge, to whom these documents were addressed, placed them in my hands; and at his request I forwarded them to Professor Sylvester, who expresses the opinion, with which I entirely agree, that the solution is thoroughly sound, and authorizes me to say that he has suggested the propriety of the publication of the papers in your Magazine. In this suggestion the Vice-Chancellor cordially concurs. I am, Gentlemen, Your obedient Servant, *The research will be found in Poggendorff's Annalen, vol. cxlvii. p.155. † Compare Pogg. Ann. vol. cxli. p. 458, wherewith it must be remarked that, after the necessary reductions, the numbers which Ammann and I have found agree very closely with the results of Waltenhofen (Pogg. Ann. vol. cxxxiii. p. 478). Oakland, Feb. 15, 1874. SIR,-In a Number of the London, Edinburgh, and Dublin Philosophical Magazine for the year 1852 or 1862, is a discussion of a simple geometrical problem by J. J. Sylvester, in which this gentleman attempts to illustrate a conjectured principle in the Theory of Geometrical Method. The problem appears to have been given out at Cambridge many years before, and to have excited the attention of some of the first mathematicians of Europe for a number of years. A direct solution of the problem was demanded; and after a protracted discussion it was concluded that none was possible. Upon that some mathematicians lent their attention, by a study of the nature of the problem under question, to discovering some rule by which on inspection they might be able to tell whether a given problem admitted of only an indirect solution, a question of some importance in astronomy. I have not been able to discover that any direct solution has been brought forth as yet; so I send you this, given to me by a friend of mine, Mr. Hesse, who solved it in 1842. I send it, thinking that every mistake cleared up in science is a step toward truth. If it would not be too much trespassing on your good nature, I should be pleased to hear of the fate of my solution. CHRISTINE CHART. In the triangle ABC, two angles, CAB and CBA, are |