VOL. LXXVIII.] PHILOSOPHICAL TKANSACTIONS. 4g7 



weio^hl of either air in the mixture be to the cubic inches of that air in the mixture; 

 and therefore .30535 the weight of a cubic inch of phlogisticated air, shall be 



to 1, as a? is to — -rrr which must be the number of cubic inches of phlogisti- 

 Gated air in 100 cubic inches of the mixture; and the weight of a cubic inch of 



Q 355 — X 



inflammable air, that is, .02613 : 1 :: 9-355 — x : ' q^q^^ the cubic inches of 

 inflammable air in 100 cubic inches of the mixture. Thus we have an expres- 

 sion for the cubic inches of each air ; these two quantities taken together are 

 equal to 100 cubic inches by supposition; from which equation is found 

 X = 7.373, the number of grains of phlogisticated air in 100 cubic inches, or 

 in 9.355 grains of the mixture; and 9.355 — 7.373 = I.982, the grains of 

 inflammable air. Now 7.373 : I.982 :: 121 :32; and the quantity of phlogisti- 

 cated air is to that of inflammable air, as 121 to 32. 



According to M. Berthollet's experiments, the quantity of phlogisticated is to 

 that of inflammable air, as 121 : 29. This is not very wide of calculation. If 

 we consider the great difficulty of obtaining these specific gravities with exactness, 

 we must be pleased to find so near a concurrence, and place more confidence in 

 experiments on the specific gravities and combinations of aeriform bodies, than 

 has generally been given them. M. Berthollet's experiments come within -J^- of 

 calculation ; and this difference will be diminished by -|, if we take the specific 

 gravities of the phlogisticated and inflammable airs in the proportion of 11 to 1, 

 as he has done, instead of Mr. Kirwan's proportion, which Dr. A. followed in this 

 calculation. 



XXIV. Some Properties of the Sum of the Divisors of Numbers. By Edward 



Waring, M. Z)., F. R. S. p. 388. 



1. Let the equation oc — I . x^ — I . x^ — i , a* — 1 . ar* — 1 x"— 1 =37* 



— px^-^ -\- qx^-^ — rx^-i -1- sx''-^ — &c. = x^ — x^-^ — x^-^ -f^ 

 a:f>~s -J- x^-7 — a?*-!^ — a^^- »s -f ar^-" -j- ^^-26 __ ^£» -35 _ ^^-4° _j_ 376-51 

 _}_ a;* - 57 — &c. . . .x^-" ± &c. = A = 0. The signs -f and — proceed alter- 

 nately by pairs unto the term x^-". The co-efficients of all the terms to tloe 

 above-mentioned {x''-") will be -f- 1, — 1 or O; they will be -f- 1 when multi- 



plied into a?*-*, where?; = —— — or = — - — , and z an even number; but 



— 1, if z be an uneven number; in all other cases they will be = O. The 

 numbers 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, &c. subtracted from h, may be col- 

 lected from the addition of the numbers 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, IJ, 6, &c. 

 which consist of two arithmetical serieses 1, 3, 5, 7, 9^ l ^i &c- 1^ 2, 3, 4, 5, 6, 7, 

 &c. intermixed. 



2. The sum of any power (w) of each of the roots in the equation a = will 



VOL. XVI. 3 S 



