FUNCTION. ~ 
‘by the nature of the inveftigation, in which fuch an expref- 
fion as 6 d xv occurs. 
Leta —-2eyt Y= 
a 
= 0; but from the original equation a = 
o, then eta / yes 
d 
vs an 
zy * 
fubfituting 
ede Zt?, ‘dy ¢ydy=0;3 
or 2y x eee —awdy +ydy =0; 
‘in which differential equation the arbitrary quantity 2 does 
Rot appear. 
In}jke manner, if the equation were +’ —2ay + +by =9, 
the firft differential equation would be rdw —ady + by 
dy = 0; the fecond differential equation 
dat —ady + byd'y + bdy’ 
from ies three equations the two arbitra Sasa aes a 
and 4 may be eliminated, and the refulting digerential saa 
poe will be of the fecond order, containmg neither 
nor 5. 
Generally, fuppofe $ (x, y) == 0, an equation between « and 
y to contain arbitrary ear a, b, c, &c. then thefe ar- 
bitrary quantities will the fame in the firlt, fecond, &c. 
differential equations eee by d (9 (x5 y)) =o,d 
(9 (a 9)) =o, 4 ( (% ¥)) 
Q (x, y)=0, and d(% (x y)) = onftant quan- 
tity as a, may be climinated, and the relulting equation will 
Now from 
oO, &e. 
be a differential equation of the firft order, between w, y, 7 
containing a conftant arbitrary quantity, lefs than the origi-. 
nal equation; again, from Q(% y) = 9, (¢ (a y)) 
s=0,d°(9 ( =, two conftant quantities a and 5 
may be eliminated, and the refulting equation will be of the 
o. 
fecond order between.«, y, dy and containing ttvo 
da’ 
conftant quantities lefs than the original or primitive a 
and fo on. 
Hence, fince a differential equation of the firft order may 
contain a co pa quantity lefs than. the primitive equation, 
ince rential equation of the fecond order ma 
contain two Seat quantities lefs than id nee 
Be. | H aoa a primitive equation i to have 
ore nt guantiy than the di Foret equation of the 
firft cele ae — 
oe ee the fecond aaa and: fo on; vad it cannot con- 
tain more, fince if you fuppofe ? (x, y) to contain thee 
more than the diffe rential eet olde ie ond order 
plain that from @ (a, y) =0,d (9% (% y)) = ©, “? 
(? (3 »)): = 0, no more than two can be eliminated, con- 
fequently one muft remain in the equation betwecn 2, y, 
dy a ge. 
a 
Th ae, conftant quantiles are satbieary: that is, 5 they are 
to be exprefled by general characters; what their valves are. 
to = muft depend on the nature of the fubjet invetti- 
- It poe not oy happen that the primitive equation 
has one more conftant quantity than the differential equation, 
two more confant quantities than the fecond differential 
equation ; only as peleeke in taking the integral, you 
aflign the primitive eerie its moft general form ; and if, 
inftead of arbitrary autos, ‘it really has only 2 — n', 
5 
BS 
eonttantr 
Having 
lytical calculation 
then- 
the procefe for determining the value of the arbitrary quan. 
tities will thew, that n’ fuch are equal to ©; hence no 
or ambiguity can arife from introducing into = primitive 
a by the procefs of integration as m nftant arbi- 
rary quantitics as it can have more than ie differential equa- 
d ad'y _2 7 
T ‘a ore 
hus fuppofer— a + aan ae +1 e 
(y a funtion of x ), then inte grating” , — iy —yta=a, 
orydy ~x dy —ydex he eer 
. . . v 
Again, integrating *- —eyt -=ax+ b, 
Now the arbitrary conftant uantities aand 4 are to be de. | 
termined by the nature of the 
a it age a when wv = 0, 
alfo , hen e putting v = 0, it as bigh that 6 mu 
ss equation is reduced to s* — + x 
vor, thayore rer aa i 2ar, oro= 
2 = Oa on ee equation Sav —~ 2ay+ y= O3 
hence no error can arife from intro ducing the pla con- 
ftant quantities @ aand é; but in gai g the gen 
of the primitive equation, they m es th be in td, 
finee if the conditions by which des are to be determined 
are nie one or both may beretained. 
s appeared t that of the firft or- 
der ie one arbitrary poy ee aay lefs than the primi- . 
tive equation ; that sabe aoe equations of the fecond order 
have two lefs, and fo on; fuppofe now that from the equa: 
m7 
none ? (a, y) =o, d (0 (* y Ys for sails at 
quan a, the other ac 
bed 
e = 0, the con )=0; d == Oy 
d ( o, eliminate a, fr (B) =o, and d (B)= o» 
ane b then the two refulting differential equation 
ht to agree with one another, and with the differential 
equation of the fecond order {d’ ( X)), obtained by eli- 
minating 2 - 4 from the pha equations @ (w, y) =o, d@ 
(»y) =O, do (a, y) = 
Hence a differential equation of the fecond order may be 
derived | paar wo differential equations of the firft order 
eachcontaining pie acral arbitrary conffant qtantity. 
Hence; if 2. a differential equation of the fecond order 
(ad ©), we find two eee equations of the firft 
order, wih fatisfy the equat »towit dX =o, 
peer an arbitrary c ce oe quantity a, 
containing an arbitrary oa quantity 4, by climinating 
from t the equations dK = o, dY =o, there will refult © 
an equation between w and y, containi ing the vo arbitrary ; 
a a and 4, and which is the primitive or integt ral 
equation of d* U = : 
— 
thus given an. cee of the a te of ava. 
we fha what manner 
Mr. Woodhoufe i pplies them ae tack invetigaton as are 
ufually performed by the method of fluxi 
Invefigation of the Prop —~ The 
curve “ies the p ie erties - 7 are = 5 be int ated, 
are not fuch as = generated by mechanical, apa 
but fach as are defined — and yi ving no exi 
ence Frain es ant of what receive arbitrary 
appointment; and c ae in deducin ice roper- | 
ties, ee is to be had to the ate tical — oc 
