THE JOURNAL 
THE ROYAL SOCIETY 
WESTERN AUSTRALIA* 
YOL. I. 
THE APPROXIMATE SUMMATION OF SERIES, IN WHICH 
EACH TERM IS A FUNCTION OF THE CORRESPONDING 
TERM OF AN ARITHMETICAL PROGRESSION. 
By 
Maurice A. Browne, B.A. 
(Read April 21st, 1914.) 
Harmonical Progressions. 
The height of a column, of air of unit cross section may be 
shown, by the integral calculus or otherwise, to be equal to 
K. log p, 
where P is the pressure at the bottom, P' the pressure 
at the top, and K a constant. But the same height may be ex- 
pressed as the sum of the heights of n short columns of air of 
equal mass. If the weight of each is w the pressures at the 
centres of the sections will be approximately P-\w t P-\w , etc., 
and by Boyle’s Law their heights will be : — 
C C 
P — P — -| w 
C 
P — (n — \)w 
where C is another constant. The sum of these heights is the 
total height. 
C 
+ 
C 
p —\w p~iw 
+ 
+ 
c 
P — (n~l)w 
= K. log 
P — nw 
or, 
P-\w 
+ 
P-\w 
+ 
+ 
P-(n~\)w 
K , 
— . log 
C 
P 
P-nw 
