V L U X IONS. 
481 
accelerating force 
of W down BA is 
f j/ 
time of describing AB varies as J -, or as 
v ac. for. 
V 
(i 
rator —o, and dividing by x-\-a, we have ixx — xx — ax^z 
o, or x — a— o, x—a. 
Ex. 14. To find the fun’s place in the ecliptic, when 
that part of the equation of time which arifes from the 
obliquity of the ecliptic is a maximum. Let AV be the 
equator, AW the ecliptic, Xv 
S the fun’s place, and SB 
_L AV ; then this part of S 
the equation of time is the 
difference of the fun’s lon¬ 
gitude AS and right afeen- ^ 
fon AB, turned into time. Sc 
Put s — cof. of the angle 
A=23° 28', x — the tangent of AS; then by Sphcr 
Trig. rad. — 1 : s :: x : tan. of AB — sx ; hence, by Plane 
X —sx - X 
P+V/X 
Px—a W 
Trig, the tangent of AS—AB 
I-}-** 1 
■•f X 
1 4 -sx 2 
min. or 
a*.vX P-v —a W—Pi X -* 2 
Px—aW — 
o ; but when a fra&ion va- 
2sWxV- 
max. or ■ 
— max. .*. its fluxion 
Pa.—a W j 2 
niflies, its numerator = o ; hence, 2Px 2 x 
F.v’a^o, or Pt 2 r2(iWr, . —p—• 
Ex. 12. To find the pofition of the planet Venus, 
when it gives the greateft quantity of light to the Earth. 
_ Let S be the Sun, E the Earth, 
V Venus: produce EV, on 
which let fall the _[_ SB, and 
with the center V deferibe 
the circular arc SA. Put 
fl=SE, b— SV=AV, x=EV, 
y— BV, then AB— b—y the 
verfed line of the angle SV A; 
and (by the Principles of 
Aflronomy) the quantity of 
light received at the earth 
from Venus varies as —- =: 
aX 1 -d-jtf*— ssxxX x 
IZllldA. llbliUAlUU'-7-- 
i+r* i-l -sx‘ 
—o ; hence, the numerator ■v-bs.v'i'— isx’ : x—o ; r— 
f 7 
sx 1 —o, and x=z*/ — — 1,04416, the tan. of 46’. 14b the 
s 
fun’s long, when this part of the equation of time is a 
maximum. 
Ex. 15. Given the bafe CB of an inclined plane AC, 
to find its altitude BA, when the time of the defeent of 
a body down the plane is the lead: poflible. Put a— 
CB, x—BA, then ^ a 2 -\-x 2 =■ y\A._ 
AC ; and (by Mechanics) the 
time down AC varies as 
a~-\-x 2 
Vx ’ 
which is therefore a minimum, or 
Cl^ -1- X ^ 
- is a minimum : hence* 
2 V.V'X x -A'X a 2 -\-X~ 
:o, or its numerator 2x 2 x—a 2 x¬ 
+x*-f 2*y, . ’. y— 
b y 
-— — max. Now by 
x 2 x 2 1 
Euclid, B. II. p, 12. a 2 —b 2 
-* 2 ,. r , , m 2 — x 2 
— 1 “ (if m 2 —a 2 — b 2 ) • 
hence, the quantity of light varies as 
2 bx—m 2 -\-x 2 
2X 
-X 2 
, which is therefore a maximum; hence, its 
fluxion 
2bx + 2xxy, 2x 3 — 6x 2 xy. 2bx — m 2 4-x 2 
4 * 6 
20, or its 
numerator 4bx 3 x+yx A x —12 bx 3 x-\- 6 ?n 2 x 2 x — $x*x—o, or 
by dividing by 2x 2 x, and uniting the like terms, we have 
—x 2 — ybx-\-yn 2 ^o , . x 2 -^-^ bxxz.yn 2 , a quadratic, from 
which *2= — 2b-\- 4/ 4^+3 nf. Hence we know the three 
Tides of the triangle ESV, to find the angle E of elonga¬ 
tion, which 2=39°. 44'. 
Ex. 13. Let Q^be an object placed beyond the prin¬ 
cipal focus F of a convex lens : to find its pofition, when 
its difiance Q^j] from its image q is the lead poflible. 
Put QV—x, FE=a; then (by the Principles of Optics) 
X-^~d . 
x : x-\-a : : x-\-a : Q q— -= a nnn. hence its fluxion 
x 
x 2 x—o, therefore x 2 —a 2 , and x—a. 
Ex. 16. Given the bafe CB, to find the perpendicular 
BA, fuch that a body defeending from A to B, and then 
delcribing BC with the velocity acquired, the time 
through AB and BC may be the leafi poflible. Put m— 
16— feet, a=CB, x— BA ; then (by Mechanics) the 
7 
time down AB — \ alfo, with the velocity acquired 
vi 
at B continued uniform, the body would deferibe 2AB, 
or 2x, in the fame time ; hence, as the fpace deferibed 
Q 
with an uniform velocity is as the time, 2x: a: : */- 
m 
a 7 1 f i 
:—X -—-aX\J — the time of deferibing BC ; hence, 
zx vi 2 mx 
fj , rr 1 <7 T —x 
the whole time = W - 4 - - a \J —= x 2 Xt/- + ~ ax 2 X 
v m 2 mx m 2 
rr 1 1 — 4 
v -—a minimum, or x -\—ax ~= min. 
ax 
- 3 _ 
2 x: 
_1 — 3 . r 
or x 2 —^ax 2 ; hence, x—-a. 
1 —1 
-x x - 
2 4 
2.v X x-\-a x -V —x X x-Jt-a* 
x* 
VOL. VII. No. 443. 
=0, and by afluming the nume. ^ x ' 
Ex. 17. Given the bafe CB of an inclined plane AC, 
to find its altitude BA, fuch that the horizontal velocity 
of a body at C after defeending down AC, may be the 
greatefi poflible. Put ^imCB, *r=BA, then CA— 
4/ a 2 -j-x 2 ■ now, (by Mechanics) the velocity at C is as 
4/x, and by the refolution of motion, ^a 2 -\-x 2 : a : : 
x which varies as the velocity at C in th.e 
6 G 
dire&ioa 
