Refohtion of attractive Powers. 195 
jS— :u t which being fubftituted for their values in the quantity 
lb x v/(x 2 +/), there will refult (f +p z + 1 fxxxz - Axis ~ 
( 1 _ i _ /) 2 1 j ir ♦ 
— Xj/Xi; — Byi;; in A and B for y and refpeclively 
fubftitute their value deduced from the given equation, and let 
the refulting quantities be A'xz and B 'yz 9 where A / is a func- 
tion of x and z 9 and B' a fun&ion of y and z ; find the fluent 
of A'xz from the fuppofition that x is only variable contained 
between the extreme values of x to a given value of z, which 
let be L z 9 then find the fluent of Li; by fuppofing z only 
variable contained between given values of z, and it will be 
the furface of the folid contained between thofe values. 
The fame may be deduced by finding the fluent of B 'yz 011 
the fuppofition that jy is the only variable quantity contained 
between the extreme values of y as before of x to a piven value 
of z 9 which let be L / z; then will the fluent of L y z contained 
between the given values of z be the furface required* 
If the folid be a cone generated by the rotation of a rectan- 
gular triangle round a fide containing the right angle as an 
axis ; hu will be a given quantity, if z be given. 
If the above-mentioned angles are given, but not right ones, 
the arc p f m and perpendicular Ah can ealily be deduced, and 
confequently the increment of the furface. 
3. To define a curve of double curvature, it is neceffary to 
have two equations expreffing the relation between the abfciife 
% and tfand their ordinates (y) given, and if the angles which 
they refpedtively make with each other be right ones ; the 
fluxion of the arc as given in the Proprietates Curvarum is 
(if -f x +y 2 )*. Find its value from the two given equations in 
terms of x 9 y 9 or z 9 multiplied into its refpedlive fluxions, and its 
fluent, properly corrected, will be the length of the arc required. 
-r 
