FUNCTION. 
Making therefore thele fubftitutions, and reftoring the value 
of ny fhall have 
(A x)’ 
ec t+Awamexe tAx JI — a), te - 
( J/(t— +). 3) + &c. which, by comparing the firft 
terms affected with A x, gives 
r=V(1—2). from which refults 
x 
dz _ I 
dx” 
The comparifon of the other terms will give the fecond 
and third, &c. differentials. 
In like manner, if 
@x = angle cof.2; 4 = = cof. (? x 
for x put x + Aw, and ga + nfor ¢ «3 then 
a+Awe =cof. (fe +2) =cof. (2). cof. 2 — fin. 
(p ~) fin. 25 and cof, ’ x) = x, and fin. (p«) = 
V(1 — cof? (px)) = Sy — x. 
Making thefe ce es and putting for fin. 2 . cof. #, 
n’ 
their values in feries, 2 — Per &c. 
n* . 
and 1 — — + &c. asin the former example; then 
2 
_ dz —@G x)* 
x + Ae a- Az Vt— #) 
(vo-#4 - | 4 &e. which gives 1 = — /(1—*«) 
dz dz I 
dz dy s1—-# 
Since therefore x being the fine of an angle, Vi— wis 
the cofine, and x being the cofine, «1 — 2* is the fine ; it 
; that the differential 
ps ae Wf of di Teta gir in ee lps ieee of 
Whe 
on ora v 
caufe of the adentity of the 
independently of the variable quantity, that is, we may affign 
to the variable quantity « any value A « whatever. 
Let ¢2 = o bea fimilar all then 
ae + Ax) = 0,a 
zc+D¢onr.Ave “yp De ga. (Ax) + Ke. = 03 
and therefore saat 
Owe = O03 
I” 
= 7 D oe= =O, &e. 
The fame equation tee fubfifts between the differen- 
tials of any order wha ever, 
Suppofe no 
ae quantities v 
then from wh < 
eaten fubfilts between all the differential equations. of any 
order whatever. 
And in general, firft and fecond differential equations, &e, 
) = 0 between two 
fome fundtion 
. 
are fo termed, ‘not only: when they are diretly deduced frona 
the primitive equation, but when they confift of any ae 
nation of the differential and primitive rue thus, w: 
the primitive equation contains ay ds e firtt differential 
equation. will contain x, y, and ~ dx J, the fecond, y, x, od : 
d’y 
ae 7 and fo on; and by what has preceded, one of thofe 
aia of equations may always be transformed into an- 
“To illuftrate the ufe of yaad equations in the transforma- 
by 
ie ne funCtions an’ example, let us take the cafe of 
and cof. the differentials of which have been 
ateady ae err 
ety = fin.x; z= cof.x; then 
dy z , 
on cof. x, and = fin. x, confequently 
a. =, and 5 =—y. 
If we multiply the firft of thefe equations by / — 1, and 
then add it to the fecond, 5 * + eae —eymev—t 
zt+yv—1 1) v— 13 
hence the equation 
zptyv 
Now from what has preceded, it appears that if p be any 
dp 
dx. ’ 
function of x, 2 is the differential of ‘the log. p; therefore 
e2+yV-—1)= “—1 + & will be the pri- 
qian on afer otek ié Grete one may be con- 
fidered as derived ; & is an arbitrary conitant quantity to be 
determined. by the a 7 the fun@tions y and z, conform- 
ably to the method of what we call finding fluents. 
For this i das ap we Bais Ge that on making x =o, we 
have oe and eof. x = 13; therefore y =0, 2 = 1. 
It is there me neceflary ‘that the equation above found 
atte thefe fuppofitions ; but in this cafe it becomes 
lo == k, and fine e equation 
fherclin will fimply | be log. es tyV—1) =x VG 
hence Jie 
Naperian log. is unity. 
yend x fubftitute their values fin. x, cof. x, and we 
obtain this remarkable formula 
cof, « + fin. ‘e9V—ise 
which from the double fignification of the radical vara — 7 
gives eq equally . 
Ts 
—Tse¢ being the number whofe 
xf 
oof, x — fin. x fae 
ai way A again combined es fufficient. to determine the 
f fin. « and cof. or by adding | them or fub- 
ie cee them we have 
é 
cof, i 
