8o Dr. Waring on 
S / Y / may be deduced, which are confequently all fundions 
• » 
of D, D 7 , D, D 7 , and invariable quantities ; and their 
fluxions PY 7 , SY, and S 7 Y 7 fundions of D, D 7 , 
• • •• • • 
D D 7 , D, and D 7 : from the funilar triangles before given 
• • 
SY : PQ — 2— PY : ~ — PO the radius of curvature 
1 Y SY 
hence PO is a function of D, D', D, D 7 , and D', if D = o; 
and from D, D 7 , SS 7 , D, D', and the point S 7 given in pofi- 
tion can be determined S 77 P, S 7/ Y // and PY' 7 ; for let S // >6 = C 
be drawn perpendicular to SS 7 = j, and S/j — />; then will S / (if 
P/ be a perpendicular from the point P to the line SS^rrrtr 
+ and S7 = and F/= y(SP= - £/*), and 
S 7/ P = y((3— S/) 2 + (CrtP/)') ; draw S 7/ Y 7/ perpendicular to 
the tangent PY, and cutting the lines SS 7 and SK parallel to 
PY in o and n refpedively ; then will oh — -- - - - ^ ^ ^ , 
S // o = C ; (and from the fimilar triangles S // oh and S ori)on~ 
l 
(h^oh) x whence S°Y U =iz*zS"o±:on^=SY will be a 
^ 'So 
known function of D, D 7 , D and D 7 , and invariable quanti- 
ties : the fame may be predicated of fimilar lines drawn to the 
PY py/ 
centers S /7/ , S 77// , &c. ; and confequently (/ x — =t/ 7 x — • 
t b I b i 
py// py • 
dtf" X ~ 77 p — / /7/ x ^TTTprt&c.) x A (where A, as before, de- 
notes the fluxion of the arc of the curve) =/xD±/ 7 xD 7 ± 
• • 
f" X D 7/ rfc f"' x D //7 rt &c. *= -vv, if v denotes the velocity; 
but as f /"-if"', &c. are fundions of D, D 7 , D 77 , D 7// , 
&c. refpedively, the fluent of the above mentioned quantity 
fD^fW+f'D' &c. can be found in terms of D, D', 
D 7 , 
