as to lamp renewals and the cost as to power, the first being equal to the price of a lamp divided by the product of its life into its candle-power, and the second to the price of one watthour multiplied by the watts per candle. So that, if L(v), C(v), and W(v) were the life in hours, the candle-power, and the watts per candle respectively expressed as functions of the pressure in volts kept constantly on the lamp, and if p were the price of a lamp and H the price of one watt-hour, then the cost per hour per candle equalled Ρ and the value of v which made this expression a minimum, they showed, was the proper P.D. to employ with the particular type of lamp. The authors gave two methods of solving this problem defined by their equation of cost-the one a graphical method, and the other an analytical one. The graphical method consisted in drawing curves from the best experimental results then available to represent L(v), C(v), and W(v) respectively in terms of v, and from these three curves finding the values of L(v), C(v), and W(v) for many values of v, then, by substitution in the cost equation, data were obtained from which a last curve showing the relation between potential difference and cost per hour per candle could be drawn. The curves on p. 45, Proc. Phys. Soc. vol. vii. 1885, showed the results obtained when this method was applied to the case of 16 C.P. 100-volt Edison lamps, using as experimental data the results of tests, made at the Finsbury Technical College, on the relations between candles, volts, and watts, and the values of the life of this type of lamp when run at different pressures, as published by M. Foussat. The curve AAA showed the cost per candle for lamp renewals during 560 hours when the price of a new lamp was five shillings; BBB the cost per candle for power during the same period, one horse-power for 560 hours being reckoned at £5, and CCC gave the total cost per candle for 560 hours. The ordinate of this latter curve had a minimum at about 101.4 volts. With the analytical method, on the other hand, empirical equations were first found to represent as nearly as possible the experimental results. Thus it was found that and 1 L(v) + C(v) =10007545 v-11-697 W(v)=37+108-007-0-07667; * Phil. Mag. tom. cit. p. 308. therefore the total cost per hour per candle equalled 100·07545 v −11·697 +H(3·7+108·007-0·07667 v). p From this equation, using the same values for the price of lamps and power as before, it was shown that to make the cost a minimum the P.D. maintained between the lampterminals should be 101·1; a result in close agreement with that obtained graphically. At the next meeting of the Physical Society a paper was read by Dr. Fleming *, in which he considered what proportion the cost of lamp renewals should bear to the cost of power in order that the total cost should be a minimum. As the result of the examination of various experiments, he showed that for any particular type of lamp the average life could be expressed as an exponential function of either the watts per candle, or of the candle-power, or of the P.D.; that is, he showed, for the lamps with which he was dealing, that where L was the life, W the watts per candle, C the candlepower, v the P.D., and A, B, α, B, and y constants for any particular type of lamp. Then, by substitution in the cost equation already referred to, he obtained the result that for maximum economy the cost of power must be to the cost of B-1 renewals as a to unity. The values for a and ẞ he gave В for Edison lamps as 64 and 41 respectively; so that he finally arrived at the result that, whatever the price of lamps, or of electrical energy, the pressure used must be such that the annual cost of renewals should be about 17 per cent. of the total annual cost. In April 1888, in a paper read before the American Institute of Electrical Engineers, Mr. Howell gave numerous examples of the application of Ayrton and Perry's graphical method to the finding of the efficiency at which various types of lamps should be run to obtain maximum economy for various prices of lamps and power. He gave a curve showing the results of his tests of Edison lamps when run at different efficiencies, but the lamps he used were so abnormally good, compared with any with which we are acquainted, that his experiments have no practical value. For example, his lamps run at 45 watts a candle lasted, he says, nearly 12,000 hours. Further, Mr. Howell's paper can hardly be said to have *Phil. Mag. May 1885, p. 368. advanced the matter beyond the point at which it was left by the considerations contained in the paper on "The Most Economical Potential Difference, &c.," read before this Society in 1885; for he took no account of the changes which more recent experiments had shown to occur in lamps during their life, and which render any results obtained by methods that disregard these changes useless for practical purposes. Numerous experimenters had found out that there was a serious diminution in the light emitted by a lamp as it grew old, and also a very considerable rise in the watts consumed per candle. Further, there was evidence to show that the shape of the curve between candle-power and time was not the same for different pressures; so that it had come to be recognized that, in order to determine the P.D. at which lamps should be run, it was useless to make a comparison merely between the efficiency of lamps when new and the number of hours they will last at various voltages. Moreover, this falling-off in the quality of a lamp as it ages led people to consider another point, viz., Might it not be possible, in consequence of this deterioration, for a lamp to be economically dead before the filament had actually broken? For the increased cost of the current required, compared with the light given out, might more than overbalance the expense of replacing the lamp with a new and, therefore, more brilliant one. Before describing the results which we have obtained regarding this interesting question of the existence of a point, called by the Americans the "smashing point," beyond which a lamp cannot be economically used, it will be well to shortly indicate the general conclusions to which previous experimenters have been led from tests on the modern glow-lamp. In November 1892 Mr. Feldman published a table compiled from the results of tests made by Prof. Thomas and Messrs. Martin and Hassler in America, by M. Haubtmann in France, and from some of his own measurements, which contained mean values for more than 500 lamps taken from 28 different factories and representing 49 different types. He divided the lamps into five groups according to their initial efficiencies, and gave for each group the average candle-power in per cent. of the initial candle-power and the average watts per candle at every hundred hours in the lives of the lamps. The curves in figs. 1 and 2 are drawn from the figures given in this table, fig. 1 showing the candle-power and fig. 2 the watts per candle as the lamps grow old. The numbers I., II., III., IV., and V. on the curves refer to the initial efficiencies of the lamps whose behaviour the curves illustrate, and, as seen from the table on fig. 1, the curves marked I. are drawn from results obtained from lamps which initially required from 2 to 2.5 watts per candle; for curves marked II. the initial consumption was 2.5 to 3 watts per candle; for curves III. from 3 to 3.5; for curves IV. from 35 to 4; and for curves V. above 4 watts per candle. These curves show that the fall in candle-power varied from 40 to 65 per cent. in 1200 hours, or, rejecting lamps having as low an initial consumption as from 2 to 2.5 watts per candle, Mr. Feldman's curves would lead us to expect that the candle-power of an average lamp would fall about per cent. in 1000 hours. 40 From the curves marked A in fig. 3, which give the results Fig. 3.-Prof. Thomas' American Tests. Watts per Candle. of Professor Thomas' tests made in 1892 on 127 lamps of 13 American makes, it can be seen that the average American lamp of that date dropped about 43 per cent. in candlepower in 1000 hours, the watts per candle in the same time rising from 42 to 69. The curves marked M on the same figure show the results obtained from the best make of lamp tested by Professor Thomas; the average of the 10 lamps tested showed a drop of 30 per cent. in candle-power in 1000 hours, and a rise from 4.8 to 6'1 in watts per candle. In September 1892 M. Haubtmann published tests on |