n = height of the cylinder inclosing the air; c = a factor which, multiplied by n, converts it into cubic millimeters; S = cubic contents of the meniscus; d = difference of level between A and B, fig. 4; = the pressure the air is under; N = the cubic contents of the gauge in millimeters; x = a fraction expressing the degree of exhaustion obtained; then x=1/([N (760/d)]/[nc - S]) It will be noticed that the measurements are independent of the actual height of the barometer, and if several readings are taken continuously, the result will not be sensibly affected by a simultaneous change of the barometer. Almost all the readings were taken at a temperature of about 20° C., and in the present state of the work corrections for temperature may be considered a superfluous refinement. _Gauge correction_.--It is necessary to apply to the results thus obtained a correction which becomes very important when high vacua are measured. It was found in an early stage of the experiments that the mercury, in the act of entering the highly exhausted gauge, gave out invariably a certain amount of air which of course was measured along with the residuum that properly belonged there; hence to obtain the true vacuum it is necessary to subtract the volume of this air from nc. By a series of experiments I ascertained that the amount of air introduced by the mercury in the acts of entering and leaving the gauge was sensibly constant for six of these single operations (or for three of these double operations), when they followed each other immediately. The correction accordingly is made as follows: the vacuum is first measured as described above, then by withdrawing all the boxes except the lowest, the mercury is allowed to fall so as nearly to empty the gauge; it is then made again to fill the gauge, and these operations are repeated until they amount in all to six; finally the volume and pressure are a second time measured. Assuming the pressure to remain constant, or that the volumes are reduced to the same pressure, v = the original volume; v' = the final volume; V' = volume of air introduced by the first entry of the mercury; V = corrected volume; then V' = (v'-v)/6 V = v - [(v'-v)/6] It will be noticed that it is assumed in this formula that the same amount of air is introduced into the gauge in the acts of entry and exit; in the act of entering in point of fact more fresh mercury is exposed to the action of the vacuum than in the act exit, which might possibly make the true gauge-correction rather larger than that given by the formula. It has been found that when the pump is in constant use the gauge-correction gradually diminishes from day to day; in other words, the air is gradually pumped out of the gauge-mercury. Thus on December 21, the amount of air entering with the mercury corresponded to an exhaustion of 1/27,308,805 .......Dec. 21. 1/38,806,688 ...... Dec. 29.