410 
w= 2y av 42 r=; 
Hence, vos. (2) 
To determin the xia co-fient of the second 
order, pe fe =P then 
SF =2lyp+e): 
As p is a function of y and z, and y is a function of 
x, therefore p is a function of z. aking now the 
paras gf eee t we have 
Oe = 2(ysh+ ph41)=0; 
d d d* 
that is, because p= 5%, and P= a 
‘wu 
a PY Pe s 
oe = oy + Fe 41)=0. (8) 
Or, since = 
a a1 
ne Recta adits» 5 (4) 
dx 
To determine the, son coefficient of the third 
erder, we may put 5% = 9P = 4, and then substi- 
F di 
tuting p for “%, equation (3) becomes 
Gam AyatP*+1)=0. 
By taking the fazions of both sides of this oe 
x an 
The quantity 42= <1 i, that to be determined ; 
(i ial ae PS ae expressed in 
Spe ae ae (2) by means of x and y: Hence 
the value of La 4 may be found. And, by a like mode 
See the fluxional coefficient of any order 
This mode of dtemining the axonal coefficients 
%, it &e. is that indicated by the analysis (art.45.) 
It evidently furnishes the follo ical rule. 
* Tet doe ths ere a spt 
4 as function of, and dividing by ds the 
rem be a new equation, which serves to deter- 
mine 92. Again, take the fluxions of the terms of this 
new equation, considering y and as functions of z, 
and the result will be an equation involving 4%, and 
% which, combined with Pepi equation, serves 
FLUXIONS. 
to determine 44, A thind equation may be formed! 
d*y dy 
from this by taking the fluxions, considering. 5% 
any 0 Rennioneel Aa earn me ns? 
former, gives the value of $s, and so on to any num- 
ber of equations whatever 
Let the fluxional coefficients of the different orders 
ree ae Gh ee 
4 = = as (2) 
Frew: die serene by again taking the fluxions, 
we find 
oy _ Cre (1 mey | ¢ 
dxt ~ as gh ona (y—ma)’ (3) 
Or, substituting for $Y its value as expressed by pate 
tion (2), and reducing, 
$b = — 1mm a o) 
ies sides of equation (3), 
conaidecing’ apie wad =? as functions of x, we shall 
have 
d Pi oP tt4 OF +R (5) > 
Where P, Gaal cee ee ee 
posed of x andy; by substituting for > Y ana &, 
their values given in equations (4) and (2), we may 
have #Y expressed in terms of « and y. 
. 47. The piers wisi ay a RRO Soar ae 
equation, 3 y—lmery LS Lihpeveitgse Bc 10 
it as ef Apts rule, Jast article, are called 
Fluzional . The equation itself is the 
Primitive eqnnton, The fluxional equation, which gives 
the value of 9 in terms of y and 2, is said ‘to be of 
the First Order. That ‘which gives the raloe oe 29 
in terms of 7, y, anid x, or else | in terms of y and x 
in id © bof He Second Order ; and so of the 
pes Diagn us, from the primitive equation, — 
yn ees kp trt's 
we have found. 
joa 
sab Kates wuateeet-satompeiealiae 
dty _ (1—m*)x dy . 
ie Gaara t Gee | 
dty 2m a? 
The f 
tive are identical. ire 
48, The equation y= may +2*— a'=0 being of the 
