Fumiga- ; 
tion, 
Functions. 
—— 
FUN 
of nitric acid as practised. by Dr Smith, who, for this 
discovery, received a premium from parliament. 
After the discovery of the oxymuriatic acid, Guyton 
Morveau, the French chemist, tried the effects of this 
gas in the hospitals.of France, with such decided suc- 
cess, as to put its efficacy in destroying the contagious 
matter beyond all doubt. The mixture which furnish- 
es the oxymuriatic acid consists of three parts of common 
salt, one part of black oxide of manganese, and two parts 
of sulphuric acid, The salt and manganese are first mix- 
ed together, and placed in vessels of stone-ware or 
glass, in the various rooms. The sulphuric acid-is to 
be added by a little at once, from time to time, obser- 
ving that the whole must not exceed the proportion 
above stated. The gas should never be evolved in a 
quantity, to excite coughing, nor to be otherwise dis- 
agreeable to the lungs. When we consider the benefi- 
cial effects of this gas, we cannot fail to see the neces- 
sity for using some of its liquid preparations for wash- 
ing the hands and other bodies employed in cases of 
contagious diseases. These may be the oxymuriate of 
lime used in bleaching, or simple water impregnated 
with the gas. (c. s.) 
FUNCHAL. See Manerra. . 
FUNCTION, in analysis, is an expression of caleu- 
lation, formed in any manner whatever from one or se- 
veral quantities on which its value depends. Thus, if 
x denote a variable quantity, and a, 6, c, d, constant 
3 ace+b 
i th 
quantities, then Chee 
z and y are variable quantities, and a and 4 constant 
quantities, the expression axy+6 y? is a function of « 
and y. For other distinctions between functions, see 
Fuuxrons, Sect. I. Art. 2. The term function was first 
introduced into analysis by John Bernoulli. 
is a function of x Again, if 
Catcutus or Functions. 
Sir Isaac Newton, the inventor of the method of flux- 
ions, made its principles depend on the properties of mo- 
tion, (see FLuxions, Art. 20—23) ; and Leibnitz found- 
ed its equivalent, the differential calculus, on the nature 
of quantities, which might be regarded as infinitely small 
in respect of others. At first, mathematicians were more 
eager to explore the rich mine which these philosophers 
had opened, than to call in question the principles which 
had led to its discovery. But when these came to be 
critically examined, it was observed, that as motion was 
an idea foreign to pure analysis, it could not legitimate- 
ly be made the foundation of one of its most important 
theories. Also, that the notion of a quantity infinitely 
little, was too vague to form the basis of a branch of the 
most precise of all the sciences. Hence it was thought 
desirable, that the calculus should have an origin purely 
analytical, and should depend entirely on the properties 
of finite quantities. 
To accomplish this reform, the late M. Lagrange at- 
tempted to model anew the principles of the calculus. 
He gave his ideas in the Berlin Memoirs for 1772, also 
in his Theorie des Fonctions Analytiques, (1797,) which, 
he says, _ contains the principles of the differential cal- 
culus, disengaged from all considerations of infinitel 
small or vanishing quantities, or of limits or fluxions ;” 
and again in his Legons sur le Calcul des Fonctions. 
In the calculus of functions, the variable quantities 
are denoted by the last letters of the alphabet z, y, &¢. 
and the constant quantities by the first letters a, 6, &¢. 
A function ofa single quantity, is expressed by placing 
the characteristic letter f or F before it. Thus fx, or 
30 
FUN 
F a, means any function.of x. To denote a function of Fun 
a quantity, that is-itself composed of a variable quanti- 
ty 2, for example 2”, or a+-6 x+¢ x, &c. the compound 
uantity is included in a parenthesis, thus /(2*), or 
fh e-heeh cx). A function of two independent varia- 
ble quantities « and y is expressed thus f (x, y); and 
so of others. a 
If two functions of two variable quantities « and 
are composed exactly in the same manner, and wi 
the same. constant quantities, for example a a? + bz +c, 
and ay?-+ by +¢, these are like functions, and may be 
expressed in the same calculation thus, fz and fy ; but 
if the constant quantities are not the same in both, they 
cannot be represented by the same characteristic in the 
same calculation.. However, if the constant quantities 
enter alike into both functions, and only differ in their 
absolute values, as in a x* and by?, these in the same 
calculation may be denoted by / (x, ayand (y, 4) 
The general notation we have in Fiuxions, art. 
18, 23, 28, 45, &c. and in art. 193, Prob. 4. is almost 
the very same as that of Lagrange. 
The theory of functions depends on the change which 
takes place’in the value of a function, when its variable 
quantity is increased by some indefinite increment, and, 
on the form of the developement of its new value. In the 
function fx = «?, when « is augmented by the quan-. . 
tity 2, then fa becomes f («+ 7) = (x 4+ 2)%= 2? 4+ 
227+, and in the function f «= «3, when 2 be-. 
comes x + i, then fa becomes f (7-4?) =(«+i)) =, 
+3 ci+3x2+4 2, and again, in the function 
fee “, when x becomes « + i, fx becomes f (« + 4) 
a 
a a. 
=i 2 2 
examination of any number of particular cases, it will 
appear that they have a common pro » which con- 
sists in the developement of f(a + 2) the new value of 
the function having always the form fx + 7p + q+. 
3 4+ &c. an expression in which the first term is fx, 
the original function, and the remaining terms are the 
successive positive integer powers of 2, the increment, 
multiplied by a series of quantities p, q, 7, &c, functions 
of x, which are entirely independent of 7, and which 
have a determinate form, that depends upon the nature 
of the original function. The truth of this analytic 
theorem, first particularly noticed by Euler, may be in- 
ferred from induction: As however it must result from 
the principles of analysis, Lagrange has endeavoured 
to demonstrate, that if the function /(« + 7) be deve« 
loped into a-series of the form 
Setip+lg+er + &. 
the terms of which consist each of a single power of 7 
multiplied by a function of 2, that is entirely indepen. 
dent of 7, the developement shall contain only the po- 
sitive integer powers of z, and cannot by any means con~ 
tain either a negative or fractional power of that quan- 
tity, provided that the value of x be altogether inde- 
terminate. If, however, particular values be given to 
“x, then the proposition will not be universally true, 
Our limits oblige us to refer to Lagrange’s work for the 
demonstration (Theorie des Fonctions), which has in 
some respects been rendered more complete by Poisson; 
Correspondence sur L’ Ecole Polytechniques, No. 3. 
It being ascertained that the developement of f(«+2) 
has in general the form 
fepippegp-Br+t &a 
in which Pg 7 &e. are new functions of «, which de- 
rive their origin from the original function fx, the next 
By an 
