Proceedmgs of the Royal Society of Edinburgh, 395 
Aug. 10. Immersion of Ju- f 11^ 13' 30" mean time at Busliey. 
piter’s Sd Sat. (11 14 51 mean time at Greenwich. 
Aug. 13. Immersion of Ju- f 13 62 27 mean time at Bushej\ 
piter’s 1st Sat. ( 13 53 48 mean time at Greenwich. 
Bushey Heath, 
Stanmoiie, Jug, IT. 1820. 
} 
Art. XXVI . — Proceedings of the Royal Society cf Edin- 
htirgh, (Continued from Vol. III. p. 183.) 
June 1. 1820. -A-N abstract of a mathematical paper by Pro- 
fessor Wallace was read. 
In the year 1808, l\Ir Wallace communicated to the Royal 
Society a paper on the Quadrature of the Conic Sections, and 
the Computation of Logarithms, which was published in the 
6th vol. of its Transactions. In that paper, general expres- 
sions were given for the reciprocal of any elliptic or hyperbolic 
sector ; also for the reciprocals of its second and third power, 
and analogous expressions were investigated for the reciprocals 
of the powers of the logarithm of a number. These were found 
by principles at once simple and elementary, without any re- 
ference to the fluxional or other equivalent calculus ; and, un- 
like the ordinary series, which in some cases converge too slow 
to be of any practical use, they are always applicable. In the 
paper to which this notice refers, the same elementary princi- 
ples are applied to the investigation of new series, for the simple 
powers of the areas of elliptic and hyperbolic sectors, and for 
the logarithm of a number, and these are at once simple and 
symmetrical in their form, and universally applicable. From 
the general expression for the area of the sector of any conic 
section, we derive the following for the arc of a circle. 
Let a denote any arc of which the rad. = 1 . 
Put V for 1 — cos. a 
n for 1 4- cos. « -f 2 cos. | a 
n' for 1 4- cos. \ a ^ 2 cos. | a 
n"' for 1 -f cos. \ a 2 cos. J a 
n'" for 1 -j- cos. J « -f- 2 cos. a 
and so on. 
