FUN 
fveetmeats: frefh meat and poultry are not. sg be had with- 
out leave of the governor, and at a high pric 
HEON, a river. oe eae Bagel of Cu Treland, 
which rifes in the Galtie pafling near the 
town o orth, waters ie ‘peautifal ee of lord Mount- 
cafhel, and nis joins the Black-water a few miles below 
Fermoy. 
FUNCTION, .the aé& sa doing fomething for which 
the agent was appointed, or to which he was obliged. 
FuNcTIon 4is ufed ignrtively in {peal king 0 of the of- 
fices, duties, or occupations in which a perfon is engaged. 
The actions of an caballdor mutt be ditlinguifhed from 
his funGtions; the one regard his character, the other. his 
perfon. 
oo in Analyfis. The function of any quantity, 
as x, is any algebraic expreffion of calculation into which x 
ene pore with © ther quantities that have invariable 
values ; thus = (1+), (14+)! (1-+5x)"log. x, &e. 
are all expreffions which may be called funCtions of x : 
here fuppofed to be a variable quantity, and the object of 
the oe of analytic functions is to inv eltigate rules and 
r determining 1 in what certain variations 
fied the aie s of the expretion, 
ner 
Though the theory of analytic fener is fo compre 
henfive as to embrace the whole doétrine of variable quan 
of what nature foever, yet it 1s no- 
thing more than a continua’ ranc 
b It requires no new h hy pothelis, but derives all i its rules 
yeti ion ‘fo often aed againtt thofe methods. Princ ples 
derived from the doctrine of bodies in motion, when applied 
ew 
] Bie ect the ae. of fome other more appropriate 
<4 
are ve it cannot be neceflary to have recourfe to mecha- 
a principles to fhew what is the ag a+ x multi- 
+ x acertain ee , yet this iscon- 
R fluxions, aa they attempt to 
demontftrate 
from the ee oF mo 
happen that we are sed in our path to the dieovery "of 
general sini ale from refleCting on the circumftances which 
n partial inftances. "This has been ie an in 
o¢trine “of fluxions. The relation between the time 
given rife to our idea of velocity ; and sii rc) 
ciples have been 
to variable quantities in gener 
method would certainly have feu firft to have ree 
blifhed the law of variable magnitudes generally, and 
then to have fhewn their application to the particular cafe 
of a body in motion. 
It was a celebrated mathematician of our own country, 
yet ee 
mmon a 
propo a. ul bfhitute for it, is not only fomewhat objection- 
- able in. - principe but very incommodious in its practical 
applica 
“new value of ¢ 
8 one 
, fabfitted between 9 x and «.° 
FUN 
In the year 1772, M. Lagrange, in the Memoirs of on 
Academy of Berlin, firft undertook to explain in what m 
ner the fluxionary or differential calculus was included in the 
theory of the sec gent ina tenes of age hee funGtions. This 
- celebrated work, en eer es Fonétions Ana- 
lytiques,”” appear aoe t has fince been illuftrated 
and extended in water hes « Legons fur le Cal- 
’ 
‘cul des FonCtions,’’ 1806. 
In thefe works the reader will find the whole theory of 
variable quantities completely de en and oe . oa 
great variety of important inveftigations. ‘Though we have 
no book in our own language which ae this fubjed on 
fo extenfive a {cale, yet the theory and Seoee ach on which 
the whole of this calculus depends, have been m ex- 
plain by R. Woodhoufe, efq. F.I =e of £ Cambridge, in his 
«Principles of eet ical Calculat: » for 
ea nefs, elegance, and penfpicuty, Ww will manne ae a long 
me remain oe rivalled, an me can nev: lled. 
It nd paeediny the lait, that 
- protent able is profetedly compo ofed. “But to enable 
ader mprehend the object propofed, we fhall e 
deaour to ‘Muibrate it by fome very ealy and fimple ex- 
m: les 
eo 
a 
upp pofe y x being a variable quantity, y is then 
aid to ae a “an tion ie fince it depends on x for its value. 
f x ar and becomes x + Ax x eX~ 
pene the ae aeaeae of 2) y will Bae a new value, ex- 
prefled in like manner by y+ A In this cafe we fee 
taal how the propofed aeration in the value of x 
will affeGt the value of y ; od poe ane x for x in the 
expreffion a x, and it becom . Therefore 
yrs ares 
aN, x 
os 4 a.Qn. 
It appears, therefore, that in the above equation, y = 
if any variation is made in the value of x, the corre (ane 
alteration in the value of y is a times as great 
feat fuppofe y = x° 
tx become x + Ax 5 + ; then 
+4 £4 = x + 33 x. A 
and Ay = 3x. Ax + 3 (2: 
Th the Felt of the eyes ae A Js or me new. value 
of y, confifts of a jingle term, namely, A x with a co-efficient. 
In the fecond example, Ay saree ye a res is vile sap Se 
ean . 
wee . + (ex x)F 
the firft of which i -efficient ; the 
hers contain regular afce ending. noes a A ky as xy he 
with anc efpe€tive co-efficien 't 
It will be ou that it is under this form that Ay, or the 
ma at e mo : — taine 
ie of refearch i ne ore ferns. and: 
fometimes the whole of this fe oon the ; Saige relation ' 
tween ¢ xan mes’ the revere ; find’ 
the dire@ and-reverfe, are ana alogous to te fi ni ing fluxidr a 
and fluents in the ufual a lonary Pesca ulus. : 
If y be any function = 0x, and its new valuey + 
be — in the form of a Aude afcénding accordin 
the ers of A xy then the firft te =f at | eries, cool 
in cient, is "alle d‘t rit’ der 7 
y 
3 
7% 
sg 
revious pate Lat arbitrary. he 
quantit n the contra arys © obtained in con fequen a 
particular coe 3 therefore, 1 iu whatever procéfs it is “appliedy 
3K2 that 
» 
