instead of x ; 
arrive at the same final 
at once #+h for x, and y-+4 
ee Sy 1 ee 
3 am a. hs 
In this st Liteon ar leis «, 
DOTS: aE 
Seeds annalaeniaian become y + 4; by which 
fis }hyg) meommen hte thy got ty the dons plotealy 
pag = ns 
oe ke bu et 
ue of y, u 
wt Tit oe ay 3 tay ast 
by cither 
result, as if we wereto substitute 
[ang Oe functions. 
of these methods, 
‘du\ 
dz} ke 
dy 2° 
Lae 
d(7*) 
But it is easy to see, that the expression dz! indi« 
cotes, chit Ua Bald of w fr Go'be taken twine ist rs- 
Lesintly,eo-4; siding ional, Vhanihdlositelyto:g $ it may 
au 
therefore be more simply wrieen thus, sae a4 
ot 
like manner, _-——+, ee Pe which means thit ‘the fluxion of 
w is to be taken once relatively to 2, and twice relative. 
ly to y, nay be written thus 7. ® Sik afciomoel 
d*+™u : A 
in geeyl, [das Om mapas Se Deennl cocci, 
ot the wih onder of the Sancta 22%, in which y 
alae decmiiierel = fon ambi geietty eal apta 
"u 
‘Ty means the fluxional coefficient of the mth order, 
hase ete tanya 
bing undertod, whet y becomes y +4, then * bes 
coms 
, i+ yeas tapas 
and ™ becomes leben 
tat ayaa! + aganza + 
and so on for, &e, ; ' : 
These values of i ap wan 8 
* To exemplify this motation, let use 2%y%, then ee saty'; a = 
= l2zy, &c. 
FLUXIONS. 
4 Peek da. 
a Sf 
a a. 
necceemaaeantt | 
ue reas 
wt tt ay et, ra ate 
Hint apa apa 
“dtu Ie ke 
tide & irre is + &. 
: a. 
+ Te a t 
“ 4p hee, 
| Thin the cmp drlpemet he new alae 
the function a. 
101. Pape pr eer a 
remain the same, 
melee +h} 
au kb Bu B ‘ 
“4G tape tapas A 
Here + enters ito the fonctions i, &e. as a con- 
stant 3 but wramntless ey 
value, and necro then « will become 
du du ht} AD sow] 
* hag t tae GUS 
Also, (employing the same mode of notation as in the 
other ten gore a. will become 
“a 
“y ye woe 
Bu he Peay ee 
. at iaay" taedy 2 2 + he 
ua age Tice ro i 
du ae 
7 + asig asap & 2 FH he. | 
and soon for > &e, Pharr 
being substituted i in 15 Saaamnaepe FS (a ER) 
“eared tos 
@u 
eet: hi ee 
Mi catriaait 2) 
a3 + 
the fmetion 
* 102. If we com in these two sore the 
nth ot , we 
terms which contain the same ee fs 
u 
> shall find this series of equations, ase Gedy’ 
Gu: ; antag a*tmu 
dyda = ardy mine Ta = Tedy 
=6 an ae Oa sh 53 = OP apt 
