ilG FUN 
confifts; and what each of its terms fignifies. He remarks 
that it is evident that, if in this fundtion we feek what is 
independent of i, w'C have only to make i—o, which re¬ 
duces y (.V + f) to Jx; thus /r is the part of f (x + i) 
which remains by making z = a; hence Jx=/ (x +i), 
added to a quantity whicli vanifheswhen fr=o, and which 
fliall be multiplied or may be fuppofed to be multiplied 
by a pofitive power of i: but it has been demonflrated 
that, in the developement of f(x-\-t}, no fradtional 
power of z can enter; therefore the quantity concerned 
can only be multiplied by fome pofitive and whole power 
of f; it will then be of the form zP, P being a funftion of 
X and i, and which fhall not be infinite wlien i~o. Af¬ 
ter fome examples, M. La Grange demonftrates this im¬ 
portant theorem, that, in the feries fx -f- pi -\-qP -j-, &c. 
2 may always betaken fufficiently fmall, fo that any term 
wliatever fhall be greater than the fum of all the fucceed- 
ing terms. 
In the fecond part of his work, M. La Grange applies 
his theory to geometry atid mechanics. The common 
and known methods in the fl uxionary and differential cal¬ 
culus for determining tangents, radii of curvature, &c. 
are excellent, if we regard their fimplicity and generality, 
but are deficient in that rigour of proof for which tiie an¬ 
cient demonfirations are celebrated. Such a deficiency 
in the theory of fl uxions is fupplied by that of functions; 
and the problems of tangents, of the curves of contact 
and ofculation, are in this work placed under a very per- 
fpicuous point of view, and demonftrated rigoroully. 
The proofs of thefe problems depend on a form fliewn in 
the firff part to belong to the developement of /(x + f) ; 
namely;- fx-\-{f'x-\- _/"x-f_ f"' j being an in- 
2 23 
determinate quantity, but contained between tlie limits of 
0 and 2;—atid on a theorem, in which it is proved that i 
may always be taken fufficiently fmall, fo that any term 
fhall be greater than the fum of all the preceding terms. 
The problems of the maxima and minima of quantities 
are demonflrated in a molt fatisfadfory manner. ’It is 
fhewn that, if the fuinStion fx ht a. maximum, or mini¬ 
mum, the prime fundtion y'x muft^o; and that f"x 
muft be •< 0 in cafe of a maximum, or >• 0, in the cafe 
of a minimum : but that, if f" x=:o,f"'x likewife rz 0, 
and /""x is <; 0 or >. e, according as the quantity is a 
maximum or minimum ; in general, that, if the derived 
fundtion of any even order difappear, the fundlion of the 
uneven order following difappears likewife, and the func¬ 
tion of the fucceeding even order is negative for the max- 
inuim or pofitive for the minimum. 
The author extends his theory to the determination of 
queflions relative to curves of double curvature, and of 
thofe problems which belong to that branch of analyfis 
called Cakul dcs .Variations, firfl introduced by John Ber- 
t.ouilli in the lolution of his famous problem of the curve 
of quickefl defcent, (brachyltochrone,) and lince per- 
fedled by Euler and the author of the work we are fpeak- 
ing of. 
In the application of the theory of fundlions to mecha¬ 
nics, the fpace is confidered as a fundlion of the time, 
and is reprefented by the equation x—ft ; hence at the 
end of a time the fpace deferibed will be f — 
ft-, or, by the form for the developement of fundlions, 
fi* fi3 
hft -}._ /"t+ f" 0 ) xbeingan unknown coef- 
2 23, . • 
ficient, the value of which is comprifed between 0 and i. 
Now in this form the firft term reprelents uniform motion, 
ft the I elation of the fpace and time being the meafure 
of that which is called velocity, the fecond term reprefenis 
motion uniformly accelerated where the quantity 1 —L.xe- 
2 
prefents the accelerating force ; the third term reprefents 
the total of the other motions; now 9 may be taken fo 
fmall that the motion compoied of the motions reprefen- 
ted by the firft two terms fi'.all approach nearer to the 
FUN 
true motion, than any other motion can, compofed of an 
uniform motion and a motion uniformly accelerated. 
From the developement of the fundlion jf (t-^ 5 ), it ap¬ 
pears that, in any redlilinear motion where the fpace de¬ 
feribed is a given fundlion of the time, the prime func¬ 
tion of this fundlion will reprefent the velocity, and the 
fecond fundlion the accelerating force in any inftant : 
whence it appears that the prime and fecond fundlions 
naturally prefent tliemfelves in mechanics, where they 
have a determinate value and fignification ; and it is this 
circumftance which induced Newton to eftablifii fluxions 
on the confideration of motion. 
It has been faid that the true principles of any inven¬ 
tion are moft fatisfadlorily and truly explained by the in¬ 
ventors themfelves.; yet it is at the diftance of more than 
an hundred years from the difeovery of the fiuxionary 
calculus, that its principles are firft clearly and rigoroufiy 
ellablifhed. It has been referved for the mathematicians 
of thefe days, to efi'edt what was denied to a Newton and 
a Leibnitz; and to fupply to tiieir theory that evidence 
and exadlnefs, the want of which makes us attribute its 
invention rather to the felicity of the times in which they 
lived, than to the excellence of their genius. See the 
article Fluxions, vol. vii. 
FUND,/, \_fond, Vx. fando, Lat. abag.] Stock ; ca¬ 
pital; that by which any expence is fupported.—He 
touches the paffions more delicately than Ovid, and per¬ 
forms all this out of his own fund, without diving into the 
arts and fciences for a fupply. Drjdca, 
Part muft be left, a fund wh.en foes invade. 
And part employ’d to roll the vvatry tide, Diydcn. 
Stock or bank of money.—As my eftate has been hithert® 
either toft upon feas, or lludluating in funds, it is now 
fixed in fubftaniial ..cres. Addi/on. 
The Public Funds t/ENGLAND, are the taxes ap¬ 
propriated by parliament for the fupport of civil govern¬ 
ment, and for the payment of the principal and inteieft of 
money borrowed for public fervices. They are formed 
into four divifionsor claffes, called the Aggz-e/rate Fund, the 
South-Sea Fund, the General Fund, -dvA the Sinking Fund. 
The aggregate fund was eftabliflied by an adl of Geo. 1 . 
c. 12. in 1715. It had this namegiven it, becaufe it con- 
fifted of a great variety of taxesand furpkiffesof taxesand 
duties which were in that year confolidated, and given as a 
fecurity for the difeharge of the iniereft and principal of 
debts due to the bank of England, and fome other public 
debt's ; and alfo forthe payment of 120,000!. per ann. to the 
civil lift. Into this fund are brought the two thirds and one 
half fubfidy of tonnage and poundage ; half the inland du¬ 
ties on tea and coffee ; the houfe-money granted by the 7th 
of Wm. III. the duty on hops ; the duties on low wines, 
brandy, and Britifti fpirits ; all arrears of land-taxes ; all 
public monies not appropriated ; the furpluffes of the 
nine-penny excife, of the five-fevenths of the bank nine- 
penny excife, of the revenues in the annuity adls of the 
4tb, jtli, and 6th, of queen Anne, &c. and by an adl of 
the ift of GeOk III. all the duties conftituting the revenue 
of the civil lift. The whole produce or income of this 
fund had been for fome years'about 2,600,000!. per ann. 
The South-Sea was eftabliflied, by flat. 3 Geo. I. c. 9, 
in 1716 ; and is fo called, becaufe appropriated to pay 
the intereft of fuch part of the national debt, as was ad¬ 
vanced by that company and its annuitants. It confifts of 
a duty on candles, and certain impofts on wines, vinegar, 
tobacco, and Eafl-India goods. 
The general fund was alfo eftabliflied, by flat. 3 Geo. I, 
C. 7. in 1716, and confifts of a fubfidy on goods exported ; 
a tax on hackney-coaches and chairs; duties on foap, 
hides, ftamps, and policies of infurance ; 700I. per week, 
letter-money ; a moiety of the inland duties on tea and 
cofi'ee ; and 39,855!. per annum out of the hereditary ex¬ 
cife on beer for bankers annuities. All thefe taxes and 
payments have for fome years amounted to a little more 
than a million per ann.-and are appropriated to the dif¬ 
eharge 
