420 
nswet part of that function, also any constant, or root 
Method of it, will be the greatest or least In some 
cases, it may be convenient to remark, that when a 
Fig. 3 
Pig. 4 
Fig. 6. 
quantity 
or decreases. 
all ri r fin te 
ee oe Mol at which has the grestest aren 
Let ABC be the triangle (Fig¢) 25 pe ae ae 
a; AB=-x; 
bp & AC= the 
— , because by 9 Pe en Be pee Fa 
het nas), "we have ya deyo—), and hence 
ao of « may be found as in the former examples. 
Or, paige eta carte and find x, so that the 
@—z*), may be a mazi- 
= (art. 64.), so that putting y/ = a? (a? — 2°), we 
ve 
4 = 2a? x—42'=0. 
Hence 2° = «from which it appears that the sides 
about the ri angle are equal n this, as in the pre- 
i nature of the question excludes 
of a minimum. 
as yd To determine the dimensions of a cylindric 
at the top, which shall contain a given 
iquor, grain, lias ass esrb 
le. 
ABCD be the measure (Fig. 4.) ; 
peace cpa the depth AB=v; the num ~— ex- 
rac circumference of a circle, of which the di- 
ity, via. 3.14159": and let ¢ be the con- 
tent of the ey 
a - Then, geometry, wa is the 
ray woes Aprons yep 
Src ofthe pine. penser Foe AN I 
== | and the solid content is =>" ; which being made 
©; this value of » being put inthe Yrq— 
+ 
expression for the concave surface, it becomes ——; 
oe the whole internal surface of ud ey/thiter i is 
Sei > which being denoted by y, we have 
c 
“rea 
8c 
=3 
Frm the it thee uty we get e#3= 
shige nee 9 eer 
we have found for x? cor- 
repo to mii ( (art. 62.) Now, since 
=8c, and x2*u=4c; we have r23=2 x 2* v, an 
hence 2=2v ; thus it appears peng ene 
base must be exactly double the depth of the measure ; 
=c, we find v= 
dy —_ 
dz” 
a 
8c, 
to be its 
the Fi 
v4 ee ee 
FLUXIONS.” 
of the distance from the point : therefore, the 
efoct of the ead to lumina te pane at A maybe Me 
expressed by “rt. Put a CAB 
in. CAB __2(1 bee a 
Ar = ; and s0, rejecting the constant 
divisor a?, we have y=x(1— 2?) =r —2*: Now we 
have already found (in Ex. 2.), this function is a 
pang ate em: dre Ape 1:.71 nearly ; there- 
fore, BC=8.5 inches, 
Ex. 7. To find the position of the Venus in 
respect of the earth and the sun, when the area of the 
peg nay tyros ep ey em 
Sun, 
Va aed VE pemepticses 
to VS and VE, The illuminated surface of Venus is. 
the céa; but of this, only the part ba 
is turned towards the Earth, Ba gto ASE oor boa 
nous crescent contained between half circumference 
of the disk and a semiellipse ; the breadth of the cres- 
Sr temas dean ev wena 
a, bei e angle a is to 
ig, tangle DVE; therefore, the breadth of the crescent 
Sres th besa tect tee ouighe ae Now, b vo 
nature of the the area of the crescent will 
iene 
nae" some 
Put “pat SVna a VE=z ; then, by eapegenin 
cos, SVE ="+2—"" Let y denote the apparent 
Wo then, from what has been shewn, we 
ve 
_ (a+? + 2a2—H) 
2ax3 , 
therefore, shige fluxions, 
a. c(3 #3 a®—4 a x—2*)_ 
Raz 
Hence we « o 
244a2=3(b*—0"), 
a quadratic which gives 
=e of bls 
8. To tion of a stral, or lb 
when it rests in ar - id one end 
upon & prop, 
E the Earth, and the Fig: 6, 
