5 
Arithmetical Progressions and Cognate Series, 
By a method analogous to that used for Formula I the summa- 
tion of other series can be effected. Instead of taking PV constant, 
as in the case of Boyle’s Law, assume the theoretical relation 
p- m y = constant. Then the heights, which are proportional to 
the volumes, may be integrated thus: — 
C(P— |w)” l + C(P-fw) m + ••• +C(P-[»-£]w) 
- f K . P m dP 
/ P' 
m 
K 
m + I 
(P(m + 1) _ P'(m + 1)) 
K 
= : , • (P(" l + l ) — [P — ww]( m + 1 )) 
m + 1 
whence, by a process similar to that employed for the harmonical 
progression, we get: — 
A m + (A + w) m + . . . + (A + [n — l~\w) m 
/ \ m + 1 2 
\A + (n—^)iv) Formula 
= (A+i>-l» w . TZ+frZ + i , III. 
\A + (re— £)w/ 
If m is put = 1 the above formula reduces to: — 
A + (A+w)+ ... +(^L + \n—Y\w) = in(2A-\- [n- l]w) , 
which is the usual formula for the exact summation of an arithmet- 
ical progression. If m = 0 each term becomes unity, and the sum is 
n exactly. But in other cases the summation is approximate only. 
Those series which are convergent may be summed to infinity* the 
general formula, derived from III, being: — 
1 
A 7,1 
4- ... ad inf. = 
1 
(m-1) . w . - 1 ) 
(A+w)" 1 
