In over-proof spirit, the per-centage o. p. always represents the quantity of water which the given spirit requires to reduce it to proof. By adding this per-centage over-proof to 100, we obtain a number which, multiplied by any number of gallons, and divided by 100, gives the exact number of proof gallons which is contained in any quantity of the spirit referred to. Thus:——A puncheon of rum gauged at 91 galls., and shown by the hydrometer to be 21 o. p., contains—— 21 o. p. of sample added to 100 121 No. of gallons of rum 91 —————— 11011 No. of gal. of proof-spirit = 11011 / 100 = 110·11 In like manner when a spirit is said to be 11 u. p., or under-proof, it means that 100 gal. of such spirit contains 11 gal. of water, and 89 gal. of ‘proof spirit.’ By deducting the per-centage under-proof from 100, we not only obtain the number of proof gal. contained in 100 gal. of such spirit, but, as in the last case, a factor which multiplied by any number of gal., and divided by 100, gives the exact number of ‘proof gallons’ contained in any quantity of the given strength. Thus:——An ullage brandy piece containing 45 gal. of spirit at 10 u. p., would have the proof value of—— Per cent. u. p. of sample 10, } subtracted from 100 } 90 No. of gall 45 ——— 4050 Quantity of proof spirit = 4050 / 100 = 40·50 Or exactly 40-1/2 gallons. The strength of absolute alcohol (sp. gr. ·7938) is estimated at 75-1/4% over-proof. It therefore contains 175-1/4% of ‘proof spirit,’ whilst proof spirit (sp. gr. ·91984) contains 57·06% of ‘absolute alcohol,’ both being by measure or volume. Thus—— (meas. of alc. × 175-1/4) / 100 = equiv. meas. of pf. spt. And—— (meas. of pf. spt. × 57·06) / 100 = equiv. meas. of abs. alc. From which we derive the ‘constant multipliers’ 1·7525 (or roughly 1-3/4), and ·5706, applicable to any number of volumes or gallons. For—— meas. of alc. × 1·7525 = equiv. meas. of pf. spt. and—— meas. of pt. spt. × ·5706 = equiv. meas. of alc. To ascertain what quantity of a spirit at any given strength is equiv. to or contains 100 lbs. of absolute alcohol, we have only to divide the constant number 2207·7 by the proof value per cent. of such spirit.[15] Thus——for a spirit 12 u. p.——this would be [Footnote 15: This number is obtained thus:—— 100 /·79381 = 12·6 (nearly), 12·6 × 175·25 = 2207·7. ] 100 - 12 = 88% of proof spirit; and—— 2207·7 / 88 = 25·1 gal. (nearly). That is, 25-1/10 gal. of such spirit would contain 100 lbs. of absolute alcohol. By removing the decimal point one place to the _right_, we have the equiv. measure of 1000 lbs. By removing it one, two, or three places to the _left_, we have it respectively for 10 lbs., 1 lb., and 1/10 lb.; from which the equiv. for all other weights may be easily obtained.