' Direct 
Method. 
FLUXIONS 
The first of these equations shews, that the fluxional coeffi- 
cient of the second order of a function, containing two vari- . 
waa sean kent a 
ively to the one, and then re~ 
other, is the very same, in whatever order we 
proceed in finding the fluxions. This isan important 
theorem in the calculus. To exemplify this property, let 
a=x* y®, then, taking the fluxion in respect of «, we find 
<* = ma" y* 5 and again, taking the fluxion in re- 
spect of yy oo = mn st yet. By proceeding 
in the contrary order, we find pane! and 
peed rans 
ia = mn) y—", the same result as before. 
' The other equations given above are merely conse- 
quences of the first. 5 
" 103. As, by the transition ofa single variable quan- 
tity from one state of magnitude to another, there ori- 
ginates from any function of that quantity a series of 
other functions, which are denominated its fluxional 
coefficients, (Art. 23. and 41.); a function of two in- 
variable ities must have an analogous 
oP single vertable aaa care 5 Sith ‘that 
tion of a single uantity (Art. 52.) wi 
of a taco of two fidepenklent variable quantities 
24h, u any function of 2, becomes w + ote +> 
+ &e. 5 
t quantities z and y, when « becomes x-+-h, 
and y becomes y 4%; then w becomes w  ~h + 
du, , @ul? , du Pu ke 
Sprit ae Nady Ot ay Tt 
ble thé flation of « is indi s 
ing its fluxiona coeficient by d 2, the Sahel fer te 
fluxion of z,) so that the fluxion of the function may 
frese the sane quantity 2, which will be expressed 
by 7. d, and the other derived from y, which must in 
ial fluxions of the function ; and, in the language 
Daumrnememnensae! Sipornire The sum of the par- 
tial fluxions, viz. T= dx + ig jis the whole fluxion 
of u considered as a ;, re: 
VOL. IX. PART I. 
the 
of 
and in the latter, u being a function of the. 
433 
When wu is a function of x only, instead of ge dx, it 
is usual to write simply du, because, when there is 
only one variable quantity, the symbol dw can have 
but one meaning; but when there are two variable 
pega it is to indicate. what part of 
the whole fluxion results from each ; which is con- 
veniently done by writing the fluxional coefficients 
du\ {du - 
thus, (7): (F ), as was done by Euler, or more sim- 
ply thus, hd 7y , as is now the common practice. 
104. As examples of functions of two independent 
variable quantities, 1. Let «=x + y, then du=da + 
dy. 
2. Let u=ary; ot =y, and da = yds, 
paz Gy d= Hay, therefore du=y det 
vdy. , 
8. Let w=, then ¥o 1 @¥_ = aug, 
y dz y dy yx dz 
dx du ady dx «dy 
== 5 dy=——;} ; hence, du= —— — 
LG soi d y yon 
yi. 
yheorsdy. In these examples, we have evidently 
got the same results as if x and y had been functions of 
some third quantity ¢, (Art. 29. and 30.) Indeed this 
ought to be the case, seeing that the fluxion of w can- 
not be affected by the circumstance of y being a func- 
tion of x, unless the particular form of the function be 
105. From the first fluxion of u, a function of the in- 
dependent variable quantities « and y, we find its se« 
r F du du 
cond fluxion thus ; because BS ae ay det, eM 
therefore ®und (Fae) +4 (Hay); but, ems 
y 7 2 #*) dat 
u u u 
* dyads dxdy;andd dy) = Tody Cte 
Tye 19 here dz and dy wre considered as constant. 
F Bu @u 
Therefore, observing that tte Sade. 
we have 
(Art.102.) 
Thus we see, that the ny Serer pind bet rN 
or- 
ee fluxions of the sec 
dest third, and higher fluxions of u, may be found 
in the same manner. 
ori funetions which contain two independent 
variab we might proceed 
tain dues Suppang’y to be a function of the three 
i variable quantities x, y, z, it. will have 
three partial fluxional coefficients, one relative to .r, 
to such as con- . 
which may be expressed by the symbol £"; another re- 
lative to y, which will be 5 and a third relative to z, 
Sr 
Direct 
Method. - 
