2l8 
Mr. Gompertz’s analysis applicable to the 
M . from x — n to x = m; that is* the first value of the 
increment M , . of the series being; M , and the last M 
x + p o n-\-p tn+p' 
or which is the same thing, the first term of the series being 
M and the last term of the series M . And thus would 
n m 
n j M 8+ . t express M K+£ -f M n+i+p + M w+£+2/) M m+Sj 
M *= M « +e +p+ M n+£+2 p 
P 
and also would „ +£ 
m -j- c 
M m+£ as is evident by writing n- f-e and m-f-e respectively in 
the room of n and m in the first of the series mentioned ; con- 
sequently, n 
m 
M 1 + „=I+. 
M ; they both being expressive 
of the sum of the same series. Moreover, because when x 
becomes n, x-\-z becomes 7Z + S, and when x becomes m, x-{-s 
X # + * 
T~ 
becomes m - f e ; therefore the symbols „ 
m 
and £ +E 
m + z 
when 
prefixed to the same function mean the same thing, that is 
X + e 
X 
J 
n 
m 
M = n + t 
m -f e 
M . 
X 
X 
Also because n 
m 
M 
p+x is the symbol for M n+p + M n+2p 
+ M m , an< ^ * M is the symbol for M m + M , b 
r m " 1 ~ ” 
P 
+ « + 2* + &c M , therefore « 
m 
M = M — M 
x n m+p 1 
