FLUXIONS. 425, 
’ —-AH+HE’—AE Circle PE aa circle P/E’p! = x(y+k)?; Direct 
_ AB CABS AE’ AH_AE; Cylinder PUG hoe yth oS mr? Methetes, 
a HE’/=¥/G x tanh, and AE= = AH x 008 hi Chlinder KE RPE’ p me + k)*h. 
therefore, after substituting and dividing by h, As the conoidal solid P'E: p/P Pee csae than 
Uaaigee “tan. h 1—cos. h one of these cylinders, and than the other, we have 
Lees i HG +. hk AH.» t=7(y+)Ph, an expression in which &’ is some 
Fig. 19. 
Suppose now i to decrease continually, then we have 
limie P—=P= Ps tmit SA a, timid BG = 15, ‘lis 
1. = 0; therefore 
ten 
dante ‘ 
tangent and curye proceed in the ‘same direction from 
‘the point of contact ; when they proceed in opposite di- 
rections, it will be [—~ pole 4 Pt, so that, including both 
cases in one formula, We tid ; 
fae Ee (A) 
This formula gives the tangent in terms of p and u, 
when the elation of p to wis known. ya also, it 
follows; that'd ¢ = == “TE so that, in addition to 
du 
the formule inv in art. 75: and art. 76. for the 
fluxion of an are; 
what has been found in this 
article, we have these 
two, 
Wa ae be (8) 
I i estigated in this 
Sects wl ond, in the , Second Part of this ar- 
we come to the Quadrature and Hectifice 
tion of Curves. .. 
Ke the phe of Solias and iy of Revolution. 
bs 
5 
ce 
a 
nag 
aE 
i 
es 
R. 
oS 
VOL. IX, PART II. 
quantity less than /, and greater than 0; and there< 
fore, 
; + == (y+). 
eters now the Huxional ratio $* for the limit 
of the ratio — i =, and observing that the limit of (y—A’)* 
gorge tee can pglledgd Dy tee 
ery 
andds=ry da. 
Hence it , that the flurion of a solid of revo» 
lution is to the product of of the general expression for 
the section of the solid, aye pla perpendicular to the 
axis, and 
e flusrion of the 
80. To find the fluxion of the surface generated by Fig. 20. 
the rotation of the curve APP’ (Fig. 20.), join the ex- 
tremities of the ordinates PQ, P’Q’ by the chord PP’, 
and draw the ts PE, P’E, meeting in E, and 
draw EF ieular to the axis AB, meeting it in 
F. By the revolution of the curve, the chord PP’, and 
the tangents PE, P’E, generate surfaces of truncated 
cones: and, by an axiom in the étry of solids, the 
surface generated by the are PP’, which may be regard. 
ed as the increment of the surface, generated by the 
curve AP, is than the conical surface generated 
by the chord PP’ , but less than the two surfaces gene- 
rated by the tangents PE, PE. Now, by mensura~ 
tion, the surface a by the chord PP’ is 
=(PQ + P'Q)PP’ () 
and the cenietacanipaneestad bi thd tangents PE, P/E 
are, taken together, 
” {rq +EF)PE + (PQ + EF) PEL (2) 
Therefore, between these two , quantities, surface 
gowenes by the are PP’ is always.contained. But as 
to P, the lines PQ; EF, P’Q’ approach 
ra so that, ultimately, PQ+ PQ, PQ + EF, 
wai EF are to be considered as aay therefore, the 
limit + the ratio of the expressions (1), { *} is, evi- 
dently the same as the limit hit) of the vatio e chord 
PP’ to PE+PE, the sum of the sangents 5 and as this 
last is a ratio of equality, (73.) it follows that the li- 
mit of the ratio of the curve surface generated by the 
are PP’, to the conical surface generated by the chord 
PP’, is a ratio. of equality. 
Put 2=AQ; i= PQ, a= arc AP, h= QQ’, the in- 
crement of x; 4 = P’H, the inerement of y; and 
(=P the increment of Also put_v for the curve 
os by the arc CP, and i for its incre. 
ment ; then we have — 
surf, gen by chord PP’= #(2y-4+4) x chord PP’, 
Hence, passing to the limits, and substituting the sur- 
i face generated by the are, for that generated by the 
and the are for the chord, also observing that 
the limit of 2y 4 his 2y, mind 
