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estimation of the value of life contingencies. 
Section I. Art. I. A function, that is an expression made 
up of certain quantities, has been often very usefully ex- 
pressed by some letter with those quantities written under- 
neath, of which that function or expression is made up. And 
frequently those quantities only are placed under, which it 
may be the particular object to bring into notice ; thus, if in 
an analysis we had the frequent occurrence of some particular 
expression// a 2 -{- x 2 -f- V x -f y, for instance, it might be con- 
venient to put some letter to represent it ; and if we had 
two or more expressions of the same form made up of diffe- 
rent letters, such for instance as ^ a 2 -}- x 2 + Vx -\-y, and 
V a* -|- z* -f- Vz w, it maybe more convenient to express 
them by some generic character, which shall still involve the 
peculiarity of each ; thus, by writing for the one M , and for 
the other M , and this or a similar mode of notation ; be- 
ar, w 
comes more necessary when we are ignorant of the form of 
the expressions to which our analysis is to be applied. 
Art. s. If for x in the expression signifying some func- 
tion of x, we write separately x=zn, x=zn-^-p, x = n zp, 
x = n-\- o,p, &c. x increasing by the continual addition of p ; 
then the sum of the terms commencing with x — n and finish- 
ing with x = m, is -f M n+/( + M n+ip &c M m ; and to 
express this operation on x, I use the symbol 
P 
n 
m 
prefixed 
to the function of x ; that is, I should write £ 
m 
M for this 
sum. This is the same as what is called the finite integral of 
