FUNCTION. 
(52) of = = aoe = 2dterm 
dv dp (# —a’)} 
d'y _ I a 
de Ve oe (# — 4); 
Example 4th.—Let y b+ (r" —(a-~ a)*\$ 
Suppofe (a — x)? = p; then y = (r' — p)! required 
the value of aed ane . J 
da 
> dwt 
dp 
ae 2%#%—2a€a 
ayy 2 it dp dy dy _ ine 
dp BW A) Fm Gap = ae = G eat 
ee es ee ee Jee 
dai? dp dae = FRE Ges 
da 
(37) = 4 (a — x) 
ay — L 22 Tar ERS ee ‘ 
ap aU p) 1 "4(e - (@—ay 
eo a ee cel! 
dw dp nf _ es 
UF 2 oo 
dx ee = = ua (7? —(a— _ yi 
ee 
(7° {a — x) Nye (r = (a—a))i ~ (F- re iF ay 
Example 5th. Let y = (x —a)*. Jax —b 
(x — a) 
g = (x — b)§ 
dy _, 49, 40 
ag de as 
(* ange (x —b)"E + (# —5)3.2 (xa) 
(x —a 
7 eee 
ay dq d°p dpd 
#7 ap. 8 4 9.82 4 (4 42) 
=(r— zw—b — 5b)#. —a)i-* 
( wa) (— ne b)%.2(*#—a) 
ale (2 — a)* 
Let t it now be ey to find the differential: of $ (p, g)» 
p.and g being funtions of x. 
4 d (p> 9) 
ing to ie powers 
d@pgt Ag), 
ce ban a= motpgta eS EDIE AD ap 
Ag) 
a ee 
= (expanding 9 (pf; ¢-+ 49), &c-) 
; d , : . 
© (5 9) SOD. apt ee “tha (Ag)* + &e. 
cae ve 
+ Geo, Ap +o feo ~Ap.Agt &e 
in which form the fymbols ere q) 8, 
dg 
the hypothefis, that ne var and. that 
a is alone variable in (py 9)3 hae putting 4 p> for 
Ag, Ap, the differential of ¢ (pf, g), or the ae affected 
a,dg = 2. dx) =e, 
? rg 
refpedtively denote the differential cmap: of 4 ( ‘pr qr 0 
(es is alon able in ¢ ( ote 
with da (fince d p = st 
do (p; 9) 
dp 
d 
dg+ dp 
If P be put for @ (p,q), then the fymbols 
aa : 
d ¢ (p, 9) q) 
dp 
confequently the 
differential of P ord P =< dg + ae d f. 
dP 
are expreffed by - ey 
-In like manner, if P denote a fundtion of f, g, ry 4, 8+ 
= 0(f,9,7,%, &c.) then its differential = ip” dp 
dP a d Po 
a ae 
The expreflions § —.d re . dg, &c. are called par- 
ir rr q 
dP 
sai ——» 
ip dg 
tial differentials of the. function P, 
partial ee co-efficien 
Lx. Let @ (p, g) or P be pein by (1 + pi)" - ef, | 
(C+ p)" 
dP =ampdg. (1 =p! ef padg.(rtp)"s 
again, let P = iy re + 7)3, 
» KC 
dP 
then ——- =2m I+ mo we 
a ae ace iy 
dP gp _ 3pg9t9¢ 
then gg = Gert 13 + Gor rays = apr reit 
ap | f+ Per | 
~ (apg + q)F 
_3P9+ 7 ay +29 ag 
° ~ (2pg+q)t tito 2 
Suppofe now y to be aes ny an equation —— 
ena a xy y 
ich cafes y is fhid ¢ to hea an ple one 
an of ty ae spe fun@tions being thofe which are undet 
the formy = ax + 62? + ex 4+ &e. Lee E] : a pars 
&c.; and let the equation on which the alae of the in eae 
function of y depends, - generally he alas by 0 (x y) 
= 0; then if y can thence be determined to be a fuion 
of ey when tien a + Aw, 9 becomes sy +D. 
tb ar 
.. Subftituting in "the equation ¢ (% y) = 0, for wand 
y the above values, we have Axwy+D.y. dex 
+ Dry. (A x)? 4+ &e. = = O;or putting ¢ (x, y) = X, and 
) ere mee, . (Ax) &e. = 0; 
expanding X + — 
and as this eae fubfitts, roe A wis, each fepa- 
rate ee ent affected with a different power of A # mu 
t= 03 
be pe 3L2 hence 
