FUNCTION. 
than can any other motion ee by an equation as v = 
2 Raed ing i 2 D*.v=D*.2r...&. D'™. 
#3 for tiapsk it other Gas fince oe 
-to be lefs than Aw 
or (D.x—D. v) At ve. 2—D+,9) a + &c. 
(1) euiehe to be < Ors — D".y) (At)? + (Deer, 
a — Drtt, y) (4 4)"4* 4 &e. (2) however {mall the 
iment At ‘be taken, which is clearly a eal except 
D.x=D.v,D’.«=D'.24, &c.andD* mA ‘2D, 
for if thefe differential co-efficients up to i no 
equal, the firft term of feries (1), mutt conti a eee 
((an"-*) of A? lefs than (A ¢)", and cont suey: | 
ee by (A ae ™, for Na — As pane i“ ad than Aw — 
ys (B" Di -) + (D" p-- —m tt -v) 
~At a &c, mut be < (D". ‘,—D a “(An™ 4+ 
(Dette —Dertvy) (A tyne + &c. which, as has been 
fhewn before, is impoffible. 
pply this ida gris to particular inftarices; let the 
t be thu sexprefled, y = a#, a being 
cafe the Ipaces are pro- 
a or D. « = a, confe- 
cee there is no other uniform motion which can 
oach ip nearly to the motion, to which correlponds 
s that to which correfponds the equation 
form ; then making y = x = at, 
oti 7 
d 
— .¢: forif poffible, let v= at, then v = » = at, and 
v= a . #, the fame equation as before. 
The conftant quantity a in the equat = af is the 
meafure of what is called the altering or) a in 
dt 
velocity of the body ii motion, or, if the body moved uni- 
the equation « = ¢ # the co-efficient may be called the 
formly with a velocity meafured by the co-efficient = ; 
it would have the ae a above fpecified, geek -_ 
the fpace defcribed by it 
all leffer values, oll differ 
(\ x), than the fpace defcribed in any other 
fiom depending on a different element, or eee (a) oe 
/ the heres 
: conftant, im 
ond example, let y = df 
vary as the 
a fec , & bei fa 
whieh motion, the fpaces defcribed from relt, 
as 
{quares.of the aiid Seb *, therefore qi = 262, 
and = b, epiequendy by what has the 
“Pix 
2.d7 
motion. defignated by the equation y = . tf is (at 
2. - 
more 
dy x , 
the point where y = x, and - wn a nearly 
the given motion (# = any other motion 
qual @ ft). than 
defignated by a fimilar equation to the eqnat iow wi =br; 
—) whence 
for alluming v =f ¢ it appears, that @ = ce 
d* x 
= “Fae . t', the fame equation as has been deduced, 
f, in which the motion of the 
= ¢ 4, is equal —— 
ar 
we Fri has the property above fpe- 
cifed ; and moreover, when the accelerating force is re- 
2 cal 
— to be calculated, it is on the co-efficient —; 
ia 
t not 
ing 
e underftood by it, t cn ‘fi ipeiication of an 
that 1s, the uniform acceleration of the ody’s motion, or 
what amounts to the fame, the continual and unifor aug~ 
mentation of the velocities, id meafure of vibe | being 
a {pace defcribed in a given tim 
Hence, in the £ a 
ence = 7," —— 
ce, inthe form Aw = ape Aet ape (A #)? 4. 
a 
ef¥ 
aa {A.8)* + &c. the whole fpace (A x) may be conceived 
compofed of a number of {paces due to particular kinds of 
; a : 
motion; the firft term ap Of is the {pace defcribed in 
d 
ae 
(A 2) is the fpace defcribed in motion uniformly er 
the quantity of the acceleration dependi "8 on the co-effi- 
aa 
the f s_* 
e fecond ae 
uniform motion with a velocity = 
2x 3 
cient = the third term ao + (Ad? is eather {pace 
due to a niotion, = nas of which has not been defcribed 
in terms as the two former have, becaufe no fuch motion 
occurs in oe: - aa. how ever, 
fame precifion as the laws of unifo mly acce- 
Jerated motion: a like obfervation is to be extended to the 
. ” e{(Sa)5 &e: 
Since the velecity (V} = . and accelerating force 
d+ r 
a = wat eee 
or2F.dt= dV, and2 F.d? ee! = V dV, whence, if 
F be given.a funtion of the time (¢) the integral of Vd V, 
() = 
“eo d V3. 
V? fete 
to wit, = may be exhibited by a funétion likewife of the 
time ; and by making ¢ a funtion of x, st = = : (3 
z 
dx ‘ 
d . . ‘ ® 
Tm reprefenting the differential co-effieients of x a funtion, 
of #, and of & a functian of x) the expreffion becomes 
3 2F.d a 
