FUNC 
‘on analytica Ai Rgomabe ts the expreffions cof. (x 
» fin. (w expanded may be ce to be included 
a the eae @ @ + w+ 
Ge (dey + &e. Ea cope hk there is no more reafon for 
confidering fin. as a fimple analytical funtion of x 
diftinct from e*, than ic or confidering e* -+ e~‘, or generally 
e+ eB + ey, KC. (0 Py vy Kee being roots of w” + 1 
= 0) as fm@ions of. aw diftin& from e' 
Qn the nature of the pie selon: lew = the 
Qa 
fine and — of an arc 4 reader bs ses aa very 
learned paper o Tranfadtions of 1802 b Wo od- 
he fe, — kei to the Treatife ri conomeley by 
Le Gend < 35 Since this article was written Mr. 
Woodhoule publithed a Treatife on Trigonometry, to 
which the reader is referred. 
The cafes which Mr. Woodhoufe next confiders in his 
prince oe es of analytical calculation are thofe of fra€tions, 
whofe numerators enominators under particular cir- 
cumftances vanifh. Let us take a fimple cafe, that of 
ve @ : ; 
-—; the fignification of this expreffion is, that x? ~ a? 
is to be palo by # x — a3 the sa of a oo be 
x + a, or ing x a+ 4, or 2a 
ever, is no dire and mer aes ee the c principles 
of calculation. | : be- 
come, when x = a3 the obvious and eal. anfwer is 
To the queftion, what does ~ 
== 3 And what would be the method of fhewing that 
a-~—-—-a - 
this value ———— was wrongly affigned? This, I ap- 
a 
— a 
(#2) (w +a), eS 
prehend; -x7 — a = 
(x= a) (x +2) 
ae (x= a), 205 but herein 
(x 
is a manifelt fallacy ; x? not generally = = (x — a) 
or inftanc EB he ae cafe of x 2: a isto be 
circum. | 
ieee effentially demanding this 
, that ~ — a be a quantity, or that x be 
greater than a 
By the inftances iar te it will more fully appear, that 
the values of vani are arbitrarily obtained, 
that is, obtained by extending a aw and obferving a certain 
order in the procefs of calculat 
ra or + wes), 
Suppofe the fraétion 7 (em a) for 
x, puta-+ (x — a), then, 
(24 + « — a)t = ak +4 ~— i = a — + &a 
_ oy EOD (cloned re 
fration = = 
. (x? — a) 
I+ (S=4)" — we. ‘ 
a or 
= (x -fajt ; inthis part of the opera. 
tion put x =a, and the expreffion is reduced to oF 
This refult is a a as 8 what the rules derived from the 
and that th 1e refult is: oo 
TION. 
and “x becomes (x ~a)E + ee &e. by. 
2 ( sm a) 
which method we could arrive at no saad cua a 
putting »« = a, the co-efficients of the terms of 
expanded, ‘be infinite yet, qi a an cue 
purpofe in view, i latter ane is as true as the former ; 
and, inftea s fubftitut (x — ie + a, but 
at oo for th fame resfon, as Q' ld be put 
(8 + Osa 8)., if it were ecuel to com- 
pute 9; by : a mial theorem 
Let now the fraGtion (F) be (2 = co ‘3 = 
r—(r 
which is the inftance given by Bernouilli, and fubfequently 
by Landen i in his cps Analyfis ; the value of the fraCtior 
(F) is required when rs 
((r? 6 + x — x°))— (rate 
ri (ri — xt) 
— =) r> — x*)? Ce 
now F — 
(rm x)E + 
ri (ri — xt 
(A the co-efficient of the third term) 
se (r? — x") 
2 (r' w)t 
ri (rt — xt) 
xk 
(rx«)}i. (rt + xd) + 3 (? + rx + x7). 
(ri + x) +4 Aer aa") &e. + &e. 
when x = =r, =ri (ri + rd) + (P+ r4 ry 
sy 
= 
=5re 
(73 + yi) = art 
Sains an ass 
Again, let the fraCtion (F) be 
an inftance given by Euler; the value of a “fradtion is 
required, when x = 43 
23(2 x? — (x? — a’))k — Zax 
now F = 3 
ai x (abs - DSS ee ey + &e.} 
— 24a3.x 
» — 
~ x—-a@ 
: 2 gt 2 
2x) (st —a?) = + A (xt a) + Ben 
a x—- @& . 
a eae ae 
hae 
a)s 
&e. = (x = a) 2 OE =; 
‘The : Gale rule for finding the value of thefe fraélions 
is as ie the differentials of the numerator and den 
nator : 7 
x — a . 2 
Thus, in F=—-~-, d (x? — a) 
x— Ga 
2xd x, 
d (x — a} 
