452 
aan f= (1 + 1) f'dv/(1—esin.* jv). 
—— = This fluent idently ona of apt ar = 
wn elie, alo bythe J 
tangent (onal (1) (2 art. 158. 
Ex. Let the civ he rol Of Avehtasicéa’ 
Pig. 27. finn) 1 n which AP = 1, the angle CAP = v, the 
PnP a; the equation of curve is2er=a, 
where a denotes & given ye In this case we appl 
pT ae (Pde + dr), (art. 76. 76) 
Fig. 34. 
a= fi rV(a+"). 
If this expression be compared with that for a parabolic 
arc, (Ex, 1..art. 157. s. it will appear that these curves 
are equal, when r is the ordinate of the parabola, and 
~ the parameter. 
159. The early writers on fluxions endeavoured, as 
much as possible, to find simple geometrical representa- 
tions of such ee call not be by finite 
‘functions. They succeeded in ems 4 
such as involved the radical ./(a + 6 x + ¢ x*), whi 
can always be made rational, by circular and h 
lic areas; when the fluxion contained a of the 
form /(a + bx 4+c«#* + dx), in some cases they could 
express the fluent by elliptic and hyperbolic arcs, and 
in others by the surface of an oblique cone. Maclau- 
pe eller dg od ert ength of thismode 
' fluents, in-his Fluzions, k ii. . 8, 
a the subject ‘was extended by D’Alembert, Mem. 
de Berlin, ae and 1748. Landen, in his Memoirs 
and Lucubrations, has arranged, i in Tables, the various 
fluents that may be found m this manner. 
pater mre’ watt sete orrar tration: in a man- 
bod te fens: and this last ma: tician, in 
sur les Nee Yranaendangen olsptiguey, and more 
recently in Exer, de Cal. In. has reduced all fluxions, in 
which the only radical is 4/(@ + b24-c2* + d2*) to 
three species ; and he has shewn how the fluents, may 
be sre by series which shall always converge 
160. "As any number of curves may be found that 
are | craggy hoe picgh or; 4 epee tems 
r We shall now resolve this 
Solution. Lach (ie st penta as Neots pian 
found, and let AQ, co-ordinates at P 
in thé curve. Draw the tangent PE, pny ellie 
AE perpendicular to the t, Put AQ=z, 
pai! agp alert ig ae Diy 
tangent t, angle EAB =u. formula (4), 
ar 77, we have in every curve whaievet (#) 
de=pdu+?, 
and s=fpdu4—5"., 
As = is to be an algebraic quantity, we maast bare Joes 
an algebraic quantity; let us suppose it =U, some 
function of u; then pdu=d U, ed put. Thus 
the relation of p to the angle u is determined. 
From E so har ne PED EEa 
co-ordinates ; then observing that PE n= EAB=u, we 
FLUXIONS. 
have Em= EA sin.u=p sin.u, Am=EA x cos. w= 
pcos. u, En=EP ocr tine <b yd wr X sin. u 
=t therefore, z= fsin.u, y=p sin.u 
scene But we found (for (A), art? 7) that 
(a4, therefore t= — 5 and 
pena d ' (3) P 
ive a complete solution of the 
blem ; for by the perdi tay = 
and obtain an equation involving 2 and y only. | We 
can also determine U and $Y in terms of 2 and y, 
and thence the value of z. Fon dicta otis held 
U=4a sin. 4u + ¢, then wa. 8 eelkiaid eu 
ts 3 du ~~ du 
—asin. }u; hence 
«= 2 acos. w.cos. u 4 asin. }-w sin, 
y= 2acos.}u sin. « —a sin. } u cos. u,. 
z= 3asin.dute 
By adding the squares of the values of z and y, and 
putting 1 for cos.* u + sin. u, we find 
a+ y= at(4 cos? fu + sin? 4 u) Ser 3 coats x), 
ate" = 003.2 bu. -— ti) 
Sith py net 2 sin. § & Cos., u for sin. 
cos.* 4.u—sin.* 3 ey Skew ie ae oh 
Ws Get 
oa cos.s $i * 
From these two last equations we readily find 
27 at xt = 4 (2% + y?—a*)3 
for the equation of the curve, which appears to be a 
line’of the 6th order: And as z=. ‘in. tu 4c, there: 
ons by bil | its value with that of «, it may be 
he truth of these woo may 
eauily be verified by putting 9 = $u, and 
that w=} cos. 9+ a cos. 39, , paeacBiaare 8, 
oar z= Sasin.g+e. ston 
uygens resolved - this his- of 
sivtlede aiiaiee (Horologiwt Oscllaioyium, Par ii) 
and Newton gave a solution u 
(Methot of . of luxione 
the attention of . Euler, 
(Nov. Comm. Acad. Petrop. dt) are has 
ven a solution, upon principles purely analyti 
fis Calcul. des Fonctions Legon, 19. The wchieotele i 
ven here agrees with his in the renglt, eakit ea bees 
DRAG FO 9 Sitter renee. 
Of the Content of Solids, 7 
161. Let A~PE p (Fig.19.)be an formed by the of the 
revolution of a Map St sl frme by the oh cn 
f=», aa PO, Fe a a of Paty int lids. 
in curve, $s content 
saafyda ee om 
We shall now apply this formula to some examples. 
