FUNC 
rence ©) = 0, or 5 dy qx. 
dx  ° dy da va. 
again -— = 0, or 
ax fy ax dy PX vay OX 
dy” dx Tay, ‘dx’ as dy? dx oP oie =e 
. dy 
ee or &e. &e 
ar , _ 
. . dX : 3 
in which forms 4 A means the partial differential co-efficient 
of X, or 2 (a, y) relatively to y, 
vx 
dwv.dy 
ferential co-efficient of X, when the differential is taken 
the partial dif- 
twice, firit relatively to w, and next relatively to y. 
Hence it follows, that when there i ation between 
two variable quantities v, y, the eq 
tween their firft differential a hai between their fecond 
differential aes ors, &c. 
wey + 2 xy -+ ms — ay 
oe sh dy + 3a'dy + ay det am eds#=o0 
dy yim 
oe ae yh we 
Again, let 9° pee ee 
3p dy —~—3axvdy— 3aydev4+ 3edr=o0 
eee 
dx 39 —34ae y—ax 
When y = i= or when y reprefents fuch a funCtion 
ofxast yy Pea fx, 
But it has been fhewn that d o Fe) ed Oa or expand- 
ing Fx. 
. dx; or fince 
re on 
Dy= s, DFx = Fe, &e. 
Fx.D.y+y.DFx=Dfx. 
fx 
Fe 
merators a denominators vanifh on giving x a particular 
value (2); putx= a,then F x =o, and the preceding 
equation | feces mes 
df x 
dix 
Suppofe now to be one of thofe fractions whofe nu- 
Dfx 
DFx’ 
the fame refult as was found above, as indeed 1 
ily be, fince each method i — from the 
&ed by a like a 
. or = 
which is 
mutt neceffari 
fame fource, and condu 
If it fo happens that D at D Fx 
x == a, then the fecond differential of the equation y F x = fx 
mi ft be taken, . fince it has been proved that F x. -D.y oy 
+y- x = 
d(Fx.bD. eG: DFx)=d(Dfsx), 
or d Fa. -y+ Fx. d(D. y) +dy- DFx +yd 
+ (DFs)=d(Dfx), 
erDF«.D.p+ Fa. D%y4+D.y.-DFx+y, D?Fx= 
, become eGo g = 
TION. 
os fo (P utting for dFa,dy, their values D Fx .dxy 
and dividing pia term ae 
ee YD. DF 
putting x = a, fince by hypot nea F Xs ‘Djad Be 
Fx = D'f-x: 
= 0, the above equation is via d to 
and confequently y = i the ne refult as already 
and the refults rail neceffarily a hai ie 
cri a rom the ‘lame ae eae and by the fam 
If x, D? Fx, puttin = 4, ‘kewile. become O» 
then the third differential of tlie equation y F x = fx 
given, 
or , and fo on. 
The following ba Soa of the cade the fine 
and cofines of angles is given by Lagrange, but the notation 
ufedby Mr. Woodhoufe is fubftituted iat ae of ae: given ib 
the *¢ Fo: nctions eae aged 
fin. (« +) = fin. x. cof.y + cof. x. fin. y 
cof, (x +) = cof. x. eof.y — fin. x. fin. y 
For x fubftitute x + Ax, and expand the functions fin. 
(++A x Ax) according to the powers 3 of 
Ax 3 de Sof Ax in thefe expanfions will be the 
derived functions or differential co-efficients required. By the 
preceding formule 
fin. (x + Ax) = fin. x. cof, Ax + cof. x. fin. — 
cof. ae of.x. cof, Ax —fin. x. fin. A 
It remains now to expand into feries the caw us fins 
Ax, col. Ax 
Affaming =. aay Sys that the for the fin. Ax 
mutt be of this for A "4 &c. mand n 
ay whole saree and tcag: ra making x andy 
be taken, and then we fhall obtain y 
te a Ax = 2 fins Ax. cof. Aw = 2 fin, Ax 
= he ean eee the fineof Ax = A (Ax)"4+ B 
Ax? + & 
fin. ero ee 2" B(A 2)" + &e. 
Alfo, fin? Ax = At (A xm + 2A B. (As) &e. 
and “W1—fine?An= Gn fats nJi= 1-4 ATS 
A.B. ox m+” __ So therefore 2 fin. Ax / t— fin. FA 
=2A x)™+ 2 Ax)? + &e. x 
&e. which is eae identical with the flow wing + 
Pa (* ay 
—_— 
which exeprefies the a e te es of 
the firft terms which ea ve rae power (Ax)” will 
give 
= 2" A; hence 
Hence it appears ae the firft term re the feries of fin. Ax 
isA wAx, ponh aii the two firft terms of the feries. 
; ~, making thefe fubftitutions 
n the cxreliosf fin. faneanse and cof. (x + Ax), and _ 
cucng ° aa e firft power of A x 
Awx)=fin, x +A. en cof.x« -+ &c. 
oe pals =cof.x—A.Ax fn. x+ &e. 
Hence the derived funGion, or differential co-efficient of 
fin. x will be A. cof, By and par of cof.x willbe — A. fin. x. 
Th ient nown — quantity, to 
be determined by the nature a the c 
aving thus found the firft differential co-effcients, a 
others may be found.in the fame manner ; 
85 
being A cof. x, D?’. fin.xwill be — A’ fin. x, and D. fin, x x 
will he — A®cof, x 
Ghee, 
