FLUXIONS. 
Direct Now, as / is altogether independent of 2’, x", &ec. y’, 
— ys Cenc ie operate 8 e-porala shee Uo wah ee =0, 
a y—1=0, 2! y"+2x" y?=0, a 
a! yf" £3 2" yy +2" y'=0, Ke. 
from which equations, we get » 
Pe eee 
ae a= 5: r 72? 
15 y's loy’y” y" 
27 = ? 
id y¥* ¥° 
97. i ig he fri re ee 
Bas Py ‘=— Cree sb + 0g a 
as the in 6 - Now, to shew 
bo ae cn faclc orentinnteh last. 
sa be sone nn ei inte 
Re a a ns 
a a 
Now, b sieameiaie pis from they 
pothetis of y, ction of 2, to, that of 2, a function of 
ro 
Ys wemust make y= » and y= — =, This. sub- 
stitution being made, Oe 
ra Ot ij? _ = (dm pay’) 
ae dyd*z 
The ea hae expensing moter, 
by substituting 5 for :', and 7 for 
98. totes 9a bsedeth hte tas 
y (a function of z) as functions of'some third quantity 
3. and in this view of the matter, neither the fluxions of "gle 
x nor y can be considered as constant. For example, 
in mechanics, we may consider x and y, the co-ordinates 
of the path of & projectile, as functiotse of /, the time of 
the motion. 
Let us suppose, that when ¢ becomes ¢4i, then z be- 
bes rota. ke lh dl and to abridge, let 
a woes by x, 2 a", &e. 
ty AY web 
fe, SE, Be. by (yo (y" wie 
Then, by Taylor’s theorem, 
y= (2) gives b=(Vh+(¥") ote 
y=F (@) givesk = yi +y" Sp ke. 
= ¢ (f) givesho rig 2” £8, 
"we first devel 
431 
Achaea of ee et a of , these three 
ie of t, 
“ > must_all. hold true at once. Therefore, by 
the value of h, as gi seo ew gwen reg 
ton in the fs, and then” putting th two values of & 
equal to each other, mas 
(y’) {ritertyec} 7 
5 oe? fa fame tro 
or = 46. 
Hence, finding the second power, of the 
sas al pig the co-efficients a0 wha of i=0, 
«(y)=y> Hr) 9°") =y"’, &e. 
therefore, (y)=4,,(y”)= £2) ge 
99. Let us again take the formula for the radius of 
curvature, as an example, which, when expressed in 
conformity to notation of | "last | ieticle, will be 
1 
me ta Si By substituting for (y) and (y”) 
ae % 
igen a and re- 
d 
* we find 
to be taken relatively 
not indeed appear in the 
to a quantity ¢, which does 
‘must be kept in view.. 
formula, but, ey it 
pete me Fluxions + Functions, which contain two Inde« 
‘ariable Quantities. 
t Variable 
Ste eilipandiaiida tcl only functions of a” 
variable quantity ; and this is the most commion ’ 
case ; but a function may involve two or more variable 
feat toa awe quite 9 dyer other, 
we su int on surface of a 
fn geet we spon ny es perpendicular to 
each eer which pass avagt i its centre, and put 2, y, 
z, for the co-ordinates of that point, and a for the radius ; 
the equation of the surface is x*-.y¢4-2*=a}, (Curve 
Lines.) Here each of the quantities x, y, 2, ‘may be 
say eapeeaap een of the other two, w may 
of one another. 
tet u be any function whatever of two independent 
le quantities x,y; or, re the notation of 
boriprs let-u=/ (x,y) ; = pnd & 
values, 80 that x 
bee 
comes and again, in conseqi 
(eh) it becomes f(2-+h, y + ees - 
supposing x , 
h, > 
y constant, an st datlteard tar asc 
peters 
ment yk for y, Gene have the complete 
rh, y+k). 
we may reverse 
