Fig. 44. 
Pig. 47. 
quantity p may also 
eliminate the tri- 
Fao capa ipo ja = 
gonometrical 
at he than scotons. 6t sh, anon. uebsia sm 
cloid, as we have demonstrated syn i 
article Epicyctom. The curve in a maiede Z Glee- 
tacaustic curve to a circle. 
of the curve 
a J(#, y, p)=9, / Q) 
that is, let some function.of 2’, and p, be supposed 
then be upposed 
arts curve HCD. 
and have. some other position H/C’ 
VOL, IX, PART U, 
is 
Let 
FL v XIONS. - 465 
‘) a : 3 AB’ =2", and B’C’= i esis on Salinabonsoh shi 
ans SADE x = ting a, other curve, and as, Tih , the two curves are 
eps. CHB Pre 1 ; exppeteed by evarap © the same form, we must 
xCB=—— 5s) ten “er 
f | BAS in CHE * pet Paretie fle’, y" pERy SOF 
Hence # = (AH+BH)= ae a4 op! tag Taylor's theorem, is equivalent 
sin. 2p sin. p h (2.) 
= ; a. (# 
m9 aap om ape x ys py 
‘By comparing this with the general formula (1), it ¢p- LOY D4+——ap hp Kitp&c. 
ep SO 8B Gus SP = tiehaé;* * The fluxion being. taken’ upon the h is that 
w Pe eases co.2p rity alone is variable, and Kh?4 &c. being t for all 
C8 AO Sereda ahem: mliplcd bya power 1+ 
= »z= a: 
“Ep cos22p’ dp cos, 2 p a SP (hig intersect each other in c, and let 
Hence, by formula (2), Ab, and c=, be the common corinne then, 
Qu cos. p  2sin. 2p sin. pation (1) olde tone of crore Boe i cane 
cmap Yoon pt ~eomrap J* HEB sion Fane pth ake ci 
oa geet pope HO dily find, in them x and y, the: co-ordinates belonging to thei 
pan ay aeysaieey Ara Pe) o common point ¢; that is, we must have 
nr 8 Ae Sf (29 P) =O oe 
and PS aah that sin. p= 2ein. 0S. p, =e 
oy hor ceeetectbacriny tae? pf Anes, alse, oy 
pe spatertanohsin ner FCI?) A= Gy h+Ki24 &c.=0, 
c=}cos.p(142sinp) a; y=sin3p¢ ; , 
From these equations it is and hence ‘we'must also have _ os 
af (x; ys P) r eis F) 
Menos yak po 0. 
Let Cand C’! be now ‘the in which 
the curyes He.D;.H’e¢ D’ touch>the curve PCC’Q, 
whose nature is requited; then, if we su h to de- 
crease continually, and at last to vanish,: C’ 
and c will approach to C, and:at last -will-coincide with 
‘it, so that x and y, which are.¢o-ordinates of c, the in- 
tersection of the two, curves. Hic'D, H’'c D’, will then 
become the co-ordinates of the curve PCQ. As all the 
terms which contain: will then vanish, we have evi 
dently this rules. i)s.»» 
Let the equation. of She, given carves, be 
S (YP) = 0% (*) 
« and y the ¢o-ordinates, and p a variable para- 
meter. From this iy the fluxion, sup- 
posing p to be variable, and all the other quantities con- 
stant, deduce this other equation, 
d @, Ys 
ob Ackiic Ba 
a ane var ea ee 
, which the nature of the curve, that 
touches all the given curves. 
This formula includes in it that of Pros. 3. 
Exampte. Let ACD, AC'D’, &c. be Parahols Se 
scribed by a projectile thrown from a en point A, 
wrish mgineds walodity tha "givin vertiod It is 
proposed to find the curve PCQ which touches them 
all, Let EF be the’axis of any one of the curves, AD 
an.ordinate to the axis, AP = a, the height due to the 
velocity of (see Prosectites), AB = 2, 
BC = y, the co-ordinates of C, any point in the curve, 
Put the parameter of the axis =p, and considering 
AD as a function of p, which isto be regarded as va- 
riable, put AD = q. 
3 
7... 
neous. 
Problems. 
—o 
