as are terminated by Planes , &c. 
3 °7 
the guiding curve is an hyperbola ; the only difference between 
that case, and the one we have just considered, being in the 
value of 7, which must be taken -f i instead of — i. 
Scholium to Cor. 3, Prop. 27, page 278. 
If the variable rectangle is given in species , and the touching 
curves are conic sections ; that is, if 
y 2 = + 7.x 2 ) , y'* = d ,z {fix + yX'), 
we shall have, for the action of the generated solid, on a point 
at its vertex by Prop. 4, 
A = 4 fx arc (tang. = “ + ( 1 + r) x (ftr + 7X 2 )) + 
4 /* arc (tang. — ^ -f ( 1 + r n ) x n (fix + 7X 2 )) — sttX, 
where r= r' = -J; and by actually taking the fluent, 
A=4 (x+ arc (tang. = + 
+ 4 (* + T^r.) arc ( tan g- = ^ 
— 27 rx, where ^c. = / 3 # 2 ( 1 -|— r~ j , j/ — 1 7^* ( 1 — J- ^ 2 )> y ,==: fl% 
(i+^ 2 ), *' = 1 + y *' 2 (1 + 
L v/^+T). 4r|3 “ 
. 4 ^ 
<P=77= 
v ( 1 -f- ya 2 ) v V 
or 
(i+y* 1 ) 
arc 
*y —v 
(sine = 
v v ,j. 
according as v is positive or negative, 
, 4 r'/ 3 «' 2 r / v ' v ' x , N 4 r'/?o.' x 
<P = ;= L -7= -f a/-, -f i), or = _ T . : — _ arc 
a/ ya ) y e- a/ — 1 / ( 1 -f ya' 2 ) 
(sine = — j as / is positive or negative. 
If, in the preceding expression, we make r and d infinite, 
and r' == o, it is reduced to 
