tavere 
Methaat 
—_—o 
458 
fluent of Mdx4-Ndy expressed by p+f( (s—S ev, 
fluent of Ndy, i pro- 
ceed in all as we have above, In ge- 
neral, we ov to begin with that term which brings 
wt aint ouh Ge ego ren 
Examrece 1. La 2b2de+50 4 4+ady=0. 
In this case M= =2bs, N= Zag +t” 
o™ 30, 0, 5 =0; as the condition expressed by 
the eatin (1) ie tied the expression 2 6 x dx +4- 
d exact fluxion. To determine 
Try) tere po : 
the fluent, we have P= Miaebe “a, = , 
S(8— ay te ( 49+ =i) 
=ay+l. feytevaa +9} 
Hence the primitive equation is 
bs pay+l jeyterV(l +} 
Ex. 2. Let the fluxional equation be 
a(rdz+ydy) dx—«dy 
y bydy=0; 
Tees to tye Tales 
a “ ax 
PERE Lalonde 
Ss Jer eee te 
These expressions for M and N satisfy the condition pe sasina qutond 
= > (we omit the calculation for the sake of 
brevity,) therefore the expression which is put = 0, is 
an exact fluxion ; to determine it, we have 
p= ardz y dz 
STEFH* Ate 
= 0 /(2t +9) + are. (tan. 
the fluents being taken upon the ms) that y is 
constant, 
dP 
N—Wy 
ay z 
dy = J@+y) FF¥ 
Jon Bienen ees 
The fluent is therefore : 
ay (ey) + are (tan. =) 4 dy + o=0, 
Ex. $. Let the fluxional equation be 
wetmt® [+ 7eemh= 
z 
M=1+4 Matty)’ 
—7y dN —ry 
dz ~(#+y)¥ 
=36y, 
Hence it appears that the first member of the equation chap, 4. 
-=0, 
FLUXIONS. 
weed Tee eerie ot ee 
76. hee ee We 
dition = 4N (which is called the condition of in- 
ility, satisfied, inquire 
ae ibe it said, we ny ext ini 
1» and’, bY 0his ich, when the expression M de + N dy 
is mali, it may become a complete fluxion. Tet 
the Huxional equation be put under this form 72 4-K=0, 
K being equivalent to This equation’ results fom 
the elimination of a constant quantity ; ¢ between a pri 
mitive equation, which ma be expressed tas e 
in enna a sane Or else ie Gg) 
minated directly, ve 
to the form Fe 9 i Aan 8 Twilleaa ee Svan 
see Example in article 49.) Put w for F (#, y), 
—_ x for the fluxion of u taken relatively to «, and 
Qdy for its fluxion reetively toy; so pre oe 
Qadyand =P + QY=0; hence $Y 4% = 0. 
parpraferetsnneridbgiey Sonne ic er 
be identical with the equation 42 2 4K= 0; therefore 
P “Pas 4 Qady, ‘ 
=z 
Q(dy+Kdzj)=du. 
Now, the second member jon comeion fluxion, there- 
pond groaning 9 aay Urry 
cists n 
fluxional euction yp RAO be alii 
e fluxion 
Let both sides of this last equation be multiplied 
U, any function whatever of u, then we have " 
meEed Ne Udu. t 
Now, U duis still a com fluxion ; ‘therefore the 
other member of the equation will also be a complete 
— and as the factor UQ may have an infinite-va- 
ety of forms, ita rb me von oe Se num- 
nfinite 
re ‘actors, such, fluxional equation be multi- 
by any one of tens it aa oA oes fen ad 
f 
ving a fluxional Sele ctacel bites 
<e ng tne ce This, 
ws of insuperable dif- 
uation is 
eae 
oy and inquires what fluxional 
rendered integrable by factors of a On 
this subject we must refer to his Instit. Cal. Int. vol. i. 
sect, 2., or Lacroix, Traite du Cal, Dif: &c. 2d’ part, 
