P. HISTORY OF ' AATHEMATICAL - School of Mathematics

252 A HISTORY OF MATHEMATICAL NOTATIONS
every interval I contains or incloses at least one point ~ of Z), I {Z}
is an interval-set which incloses Z broadly (i.e., not necessarily
narrowly), Fl(x) has the value 0 or F(x) according as the point x of the
interval ab lies or does not lie on I.
629. Residual calculus.-In various papers Cauchy developed and
applied a calcul des reeidue which bears a certain analogy to the
infinitesimal calculus. If f(x) =00 for X=Xh and ,(X) = (x-xl)f(x) ,
then f(x) = ((x) when X=Xl, and ((Xl) is called the residue of f(x)
X-Xl
with respect to Xl. Cauchy represents the operation of finding the
residue by a special symbol. He says:' "Nous indiquerons cette extraction
a l'aide de la lettre initiale &, qui sera consideree comme
une nouvelle caracteristique, et, pour exprimer le residu integral de
f(x), nous placerons la lettre & devant la fonction entouree de
doubles parentheses, ainsi qu'il suit: &((f(x)))." Accordingly,
& ((~~~)))
stands for the sum of residues with respect to the roots of
F(x) =0 only, while &(~~:~))
f(x) relative' to f(~) = 0 only. Furthermore.t
stands for the sum of the residues of
where z=x+yv=i represents the residue of fez) taken between the
limits X=Xo and x=X, y=Yo and y= Y. Laurentt employs the notation
&cf(z), wheref(c) = 00, or simply &f(z) when no ambiguity arises.
B. Peirce! followed in the main Cauchy's notation. D. F. Gregory"
changes the fundamental symbol to the inverted numeral 3, viz.,
l:, and distinguishes between the integral residues and the partial
residues by suffixing the root to the partial symbol. Thus &.f(x) ,
l:bf(x) are partial, Rf(x) is integral.
1 A. L. Cauchy, Exercices de mathbnatiques (Paris, 1826); (EUllTes completes
(2d ser.), Vol. VI, p. 26.
2 Cauchy, op, cit., Vol. VI, p. 256.
a H. Laurent, Traite d'analyse (Paris), Vol. III (1888), p. 243.
• Benjamin Peirce, Curves, Functions, and Forces, Vol. II (Boston, 1846), p.
43-59.
6 D. F. Gregory, Mathematical Writings (Cambridge, 1865), p. 73-86; Cambridge
Mathematical Journal, Vol. I (1839), p. 145.