function 
the coefficients of tin- snn-rssive terms the smvcssi\ < 
valin-s of a ilisrn-ti- t'nnrtiMi. Tims, <' is the 
function of 
t+jjt" rtc, 
Goniometric function, one of the six quotients of two 
si. Irs of an oblique triangle rnnsidered as a function of two 
of the angles. Graphometrtc function. See '/>/". 
metric. - Gudermannian function. *vvGuui-ri,i,initin,i. 
Hamiltonian functions, a series of functions Intro- 
ilm-ed into dynamics by Sir William K. Hamilton, any one 
of which may be used instead of the Lagrangian function. 
The common Hamiltonian fnnctinn expresses the sum of 
the kinetic and positional energy. Hankel's function, 
the function 
L'408 
in the development of (1 2x > a~) l /i according to as- 
.ending powers of n.- Polydromic or polytropic func- 
tion, onr wlurh is not nionotropic. Polyhedral func- 
tion. See i,lali,;/r,il. Potential function, the fune- 
lion expressing the potential of attractions upon a parti- 
" function, the time-integral of the l.a 
<-le. Principal 
^r:ini:i;ili I'lUU'tJi 
-ii'-li that 
Qn function, :i hiirinonir function 
. 
where s >1, and where $y=0 for y = o, y=l, y = 1, wliile 
<|>y = l for all other values of the vaviahle. Harmon- 
ic, holomorphlc, etc., function. See the adjectives. 
Heine's function, the function 
O(x, a) = c n n [(1 e2)/(l e2<n-r)). 
Homogeneous function, an algebraic polyuoiuial in 
two variables, all the terms being of the same degree. 
Hyperabelian function. See hiiperabelian Hy- 
perbolic function, (a) A Oudermanniaii function, (ft) 
One of several functions related to y 1 + k'- Binh- $ in the 
same manner In which ordinary elliptic functions are re- 
lated to i/ 1 k" in-' 4, heinn merely transformed elliptic 
functions. Hyperdlstributive, hyperelllptic, hyper- 
fuchsian, hyperspherical, etc.. function. See the ad- 
jectives. icosahedral function, see polyhedral. 
Illegitimate function, one which follows one law for 
some values of the variables and another for others. 
Implicit function, one which is defined by an equation 
of which the function does not form one member. Tn- 
tegrable function, a function such that, If the integral 
Iwtween two values of the variable be divided into infini- 
tesimal parts, and each of these be multiplied by the maxi- 
mum value of the function, then the sum of the products 
has a determinate value irrespective of the mode of sepa- 
ration of the interval into infinitesimal parts, so that the 
function has a determinate integral. Integral func- 
tion, a holomorphlc function : hut with some writers an 
algebraic )>olynomial is meant. See entire function. 
Intermediary function. See intermediary. Interpo- 
lary function, a kind of function used in interpolation. 
Irrational function, a function which cannot l>e ex- 
pressed as the ratio of two algebraic {lolynomials in Its 
variables. Irreducible function, a function u con- 
nected with its variables, 3, </, etc., by an equation F (*. 
etc., it) = 0, which cannot be separated into indepen- 
dent factors. For example, ?/ =3 y i is an irreducible func- 
tion, for (y-' z) = can be separated only into the fac- 
tors (y + y x) (yji). which have no general meaning 
independent of each other. If the Riemann's surface of 
an irreducible function consists of several sheets, these are 
all connected : and Uii- may be taken as the definition. 
Irreproductive function, a reproductive function of 
order zero. Iterative function. See itemtitr. Ja- 
cobian function, one of the functions e, H, etc., em- 
ployed by Jacobi as subsidiary to the study of elliptic 
functions. J function, the Hesselian function of the first 
kind. Keplerian function, a function expressed by an 
equation similar to that of Kepler's problem. - - Lacunary 
function. See laennarn. Lagrangian function, the 
kinetic diminished by the positional energy, or by what 
corresponds to the positional energy in the ease of varia- 
ble forces. Lamp's function, a kind of Laplace's func 
thin in which the three direction cosines enter instead of 
the radius vector, latitude, and longitude. Laplace's 
function, spherical function, or phe.rical harmonic, a 
function of two variables analogous to a trigonometrical 
series, used to express the distribution of any continuous 
quantity over a surface. A Laplace's function of the nth 
order is any function Yn of the two variables ^ and , 
which satisfies the differential equation 
+ n (n + 1) Y n = 0. 
See equation of Ijaplaccx functions, under equation. - Le- 
gendrian function, one of the xn functions of spherical 
harmonics. Limited function, one which has a maxi 
mum and a minimum value within some finite interval of 
the variable. Longimetric function. See limginietric. 
-Major function, a certain function used in the theory 
of Abelian functions. Meromorphic, metabatic, mod- 
ular, monodromlc or monotropic, monogenous, 
monotonous, multiform function. See the adjectives. 
- Non-uniform function. Same as , uniform fu iictinn. 
Normal function, a spherical harmonic of a higher 
order. Numerical generating function, the generat- 
ing function showing the number of asyzygetie invariants 
of each degorder. Octahedral function. See polyhe- 
dral. Odd function, one which changes its sign with the 
variable. One- valued function, one which has only one 
value for each set of values of the variables. Order of a 
function, the order of the algebraic differential equation 
of lowest order which connects the function with its varia- 
ble. Ordinary function, a differentiable function which 
in reference to no axis of abscissas possesses an infinite 
number of maxima. Partitively continuous, differ- 
entiable, etc., function, a function such that the inter- 
val of the variable considered may be so divided into parts 
that the function is continuous, differentiate, etc., in each 
part. Periodic function, (a) As ordinarily understood, 
a function which, whenever the variable is increased by a 
certain constant, called the period, has its value unchanged. 
(6) In a generalized sense, a function which has its value 
unchanged by the substitution for its variable of a certain 
algebraic function thereof. A periodic function of the 
second kind is one for which this function is linear. - 
Perturbatlve function, see pertur&otiw. Hord'i 
functions, hypergeometrical functions of two variables. 
Plane or planimetric function, a function expressing 
one of the relations between the areas of the three trian- 
gles formed by joining a variable point in a plane to the 
vertices of a fundamental triangle. Pn function, the 
Legendre's coefficient of the nth order, the coefficient of a n 
D M (1 - M 2) Dm Y -r j-n I 
1 (>/ - x) = n (in t l)tjn(.") I',, (.n. 
Quasi-periodic function, a function which returns to its 
value multiplied by a constant when the variable is in 
neased by a certain constant called the quasi-period. 
Radical function, a rational, integral, and homogeneous 
r\pi -i-ssion in Abelian functions having one characteristic. 
Rational and Integral function. Sec entire func 
Hen. Rational function, a function whose value in 
ten i is of the variable is expressible as a rational fraction. 
Reciprocal functions, a pair of functions f and f i, so 
related to each other that if / is one of the values of tx, 
then x is one of the values of f ly, and conversely. Each 
function is also said to be the reciprocal of the other. The 
term converge would lie preferable. Representative 
function. See reiiretrntatim. -Reproductive func- 
tion of order n, a function such that, for a certain con- 
stant c, the equation hold* f( rx) = c n f(x). - Riemann's 
function, a function satisfying the differential equation 
of the hypergeometrical series, and denned by Riemann by 
means of the properties of its critical points. It is denoted 
by P. Rosenhaln's function, an ultra-elliptic function 
of the first kind. Scalar function, a real numerical 
quantity having one or more values for each point of three- 
dimensional space. -- Sigma function. See nifjtna. Sim- 
ilar functions, (a) Functions which admit the same 
substitutions. (6) Two physical quantities whose several 
mathematical relations to two other physical quantities 
are the same. Sinusoidal function, a simple harmonic. 
Spherical function. See iMiJum't function. ste- 
reometric function, a ratio of two of the tetrahedrons 
formed by joining a variable point in space to the four 
summits of a fixed tetrahedron.- Striped function, a 
function which is represented by a pattern In stripes.- - 
Sturmian function. See Murmian. Supposition- 
less function, a function subject to no general condition 
whatever which may, for instance, be either limited or 
unlimited. Symmetric function, a function of several 
variables whose value is never altered by interchanging 
the values of any two of the variables.-- Synectic func- 
tion. Seei/n:ti'<:. Tetrahedral function. See poly- 
hedral. Theory of functions, a branch of mathematics 
which concents the general properties of different general 
forms of functions. It is sometimes regarded as embra- 
cing the entire theory of the higher functions, such as the 
gamma function, spherical harmonics, elliptic functions, 
etc. Thermodynamic function, the amount of heat 
which a body will give out in being brought to a standard 
pressure and temperature. Theta function. See theta. 
Toroidal function, a function serving to express the 
potential of an anchor-ring.- Transcendental func- 
tion, any function not algebraic.- Trigonometrical 
functions. See trigonometrical. Uniform function, a 
function such that its variable, while remaining within 
given limits, cannot pass through a cycle of values so as 
to return to its original value without the function also 
returning to its original value. Unlimited function, a 
function which within every interval has values greater 
than any predesignate finite limit and other values less 
than any predesignate finite limit. For example, suppose 
that if = when x is irrational, while t/ = ( l)i>q when y 
is equal to the irreducible fraction ;)/</. Then, although 
t/ never becomes infinite, yet between any two assignable 
values of x it has values greater than any predesignate 
positive numlwr, and values less than any predesignate 
negative nllmlwr. Vector function, a quantity of the 
nature of a vector, having magnitude and direction, dis 
tributed through space so as to have a definite magnitude 
and direction at each point. Velocity function, in tit/- 
dndyiMMMfM, a scalar function whose partial differen- 
tial coefficient for a linear displacement of the. variable 
point is equal to the component velocity of the fluid in 
that direction at that point. Vital functions, functions 
immediately necessary to life, as those of the brain, heart, 
and lungs. Weierstrassian function. See HViVrro- 
>'*</. Xn function, a Legendrian polynomial of the nth 
order, or function of the latitude and longitude on a 
sphere, satisfying Laplace's equation. Yn function, the 
Laplace's nth coefficient, lieing what Pn becomes when 
for the variable x we substitute x = cos 6 cos f\ -- sin ft 
sin i_cos(*-*]). zeta function. See uta. 
function (fungk'shon), r. t. [< function, n."] 
To porf orm a function ; work ; act ; function- 
ate ; especially, in physio!., to have a function ; 
do or be something physiologically. 
It seems probable that the policy here given formed 
the ground of an action in the Insurance Court created 
by the statute of Elizabeth, . . . which functioned . . . 
till towards the end of the seventeenth century. 
F. Martin, Hist, of Lloyd's, p. 48. 
The endodermal sac forms the axis of the tentaculocyst. 
its cells secrete crystalline concretions, and it function* 
as an otocyst. E. R. Lanketter, Encyc. Brit., XII. 551. 
functional (fungk'shon-al), a. [< ML. functio- 
n/ilia, < L. functio(n-^, function: see function, 
.] 1. Pertaining to functions; relating to 
some office or function. 
Myopy is a structural defect : presbyopy is & functional 
defect. Le Conte, Sight, p. 50. 
2. Pertaining to an algebraical operation : as, 
a functional symbol. 3. Having the function 
usual to the part or organ : as, functional wings 
of an insect (that is, those used for flying). 
Functional determinant, disease, equation, etc. See 
the nouns. 
functionality (fungk-shon-al'j-ti), n. [</(- 
tional + -ity.] The state of having or being a 
function. 
fund 
This peripheral area, which possess*^ a known and in. 
dliputable/uncfionoiKy. 
Tr. for Mi.-, i. ,i,,,l .\ .,.,/., \ 111. 170. 
Fum-li<nii,lit>i. in Analysis, is dependence on a variable 
or variables. ' Knriif. Brit., IX. 818. 
functionalize (fungk'tshon-al-iz), v. t.; pret. 
and pp. fitiii-tiinnili^t'il. ppr. fiuictioiitili^iiig. [< 
I'liiii-tioiidt + -i:e.~\ To place in a function or 
office ; assign sfmic function or office to. L<iin</. 
[Rare.] 
functionally (fungk'shon-al-i), adv. In a func- 
tional manner; bv means of functions; specifi- 
cally, in ?<)/., with reference to function alone: 
as, the maxillie of crustaceans are inorphologi- , 
(ally limbs, but functionally jaws. 
The elytra of a beetle and the halteres of a tly, though 
morphologically wings, are notfiiiietiniiallii so. Huxley. 
y-'ii/i^iimoW// produced modifications have respectively 
furthered or hindered survival in posterity. 
//. x/H-ncer, Bata of Ethics, 5 69. 
functionary (fungk'shon-a-ri), . ; pi. fitnctioit- 
in-ien (-riz). [= P. fonctioniiairc = Sp. fuiicin- 
inirin = Pg. funccionario, < L. as if 'functiona- 
riiix, < functio(n-), function: see function, .] 
One who holds an office or a trust : as, a public 
functionary ; secular functionaries. 
Their repuhliek is to have a first fiuii-tioiiani (as they 
call him), under the name of king, or not, as they think 
tit. Bvrlre, Thoughts on French Affairs. 
functionate (fungk'shon-at), r. i.; pret. and pp. 
t'li/irtioHatetl, ppr. functionating, [(.function + 
-te 2 .] To act ; have or fulfil a function ; func- 
tion. 
Tims an image is formed upon the retina, the optic nerve 
transmits the excitation to its ganglion, this at once/ur- 
liimatrx, the force called perception is evolved, and the 
image is perceived. /'"/*. ,sW. Mo., XXXI. 8. 
functionize (fungk'shon-iz), i: i. ; pret. and pp. 
functi<nti:cd, ppr. functionisino. [(.function + 
-i>e.] To function. [Rare.] 
A soul that is self-conscious is not so singular as a brain 
fnnctitniiziit<t aliout itself and its own being. 
S. Pm-trr, Human Intellect, f 41. 
functionless (fungk'shon-leg), a. [< function 
+ -fexs.] Without function or office. 
The os coccyx in man, though function!*** as a tail, 
plainly represents this part in other vertebrate animals. 
Darmn, Descent of Man, I. 28. 
Adult whales have . . . fmictiunleii rudiments of hind 
limbs imbedded in their flesh. 
Contemporary Ren., LI. 675. 
functusofficioffungk'tus o-fish'i-6). [L.:/>ic- 
tim, pp. of fungi, perform ; offieio, abl. of ofli- 
cium, duty, office.] Having performed to the 
end one's official duty; having fulfilled a func- 
tion or retired from an office. In law, "an ex- 
pression applied to an agent or donee of an authority who 
has performed the act authorized, so that the authority is 
exhausted and at an end." Rapaljfi and Laitveiice, Law- 
Diet. 
fund 1 (fund), . [In lit. sense also fond (see 
fond*),fun(l being accom. to the L. form ; < OF. 
fond, a bottom, floor, ground, foundation, also 
a merchant's stock or capital, F. fond, bottom, 
ground, fonds, estate, pi. foudg, funds, stock, = 
Pr. font = Sp. fondo, fuiido = Pg. fundo = It. 
fondo, < L. fundus, bottom, also, in particular, 
a piece of land, a farm, estate, orig. *fiiduun = 
E. bottom : see bottom. Hence (from L. fundtts) 
ult. E. found*, foundation, etc.] If. Bottom. 
See in the fund, below. 2. A stock or accu- 
mulation of money or other forms of wealth de- 
voted to or available for some purpose, as for 
the carrying on of some business or enterprise, 
or for the support and maintenance of an in- 
stitution, a family, or a person : as, a sinking- 
fund; the funds of a bank or corporation; the 
Widows' and Orphans' Fund, etc. A fund may be 
either nrtire or passim. It is actier when the bulk of it 
is invested in the subjects of the business or enterprise, as 
merchandise, ships, factories, land, bank-loans, etc. ; pas- 
xivf when it is invested in such a way (as in real estate or 
.stocks) as to produce a fixed or nearly uniform income, 
which alone is used for the specific purpose, or when it is 
used or drawn upon directly for expenses, being insuffi- 
cient to produce the requisite income by investment, or 
when it is maintained by collections or contributions for 
-I ,.'< iiir objects, as the support of missionaries or of chari- 
table enterprises. Both active and passive funds may be 
either indiridnal or collective; when collective, an indi- 
vidual interest in the former usually consists of a partner- 
ship or the ownership of joint stock, and in the latter of 
membership or of some right of joint control, unless the 
contributions are absolute gifts. 
The parliament went on slowly in fixing the fund for 
the supplies they had voted. 
Up. R-untft. Hist. Own Times, an. 169S. 
3. A store of anything to be drawn upon at 
pleasure; a stock or main source of supply; 
especially, an equipment of specific mental re- 
sources; a stock of knowledge or mental en- 
dowment of any kind : as, &fund of wisdom or 
good sense; a fund of anecdote. 
