: : 4,? 
_  ([p+2ck pt+2ck\ 2 b?(pd-+ed*) —b*(pk+-ck*) +2aby/ (pd+ ed? )sin.( — #) y-La®sin. (—w) 2x2 : 
yao (() + ob? 7 
[d -k&+cos.(—«)x]*? ; which after reduction becomes y= 
2cd\* ab d 2\sin.( —w fain. ( — pW S\" . 
(Po (AE) Piso: dh a Bad sol )xra*sin. (~ w)?x ) —[d—k-+co0s.(—w) x]? ... (15)5-an 
equation which characterizes the sections of a solid formed by the revolution of the conic sections about an axis per- 
pendicular to their principal diameters. In the case of the revolving curve being a parabola, c=0, and a*—(a?+¢b*) 
sin.p? =0=n ; which constitutes the failing case remarked above. Assigning the same values to the coefficients as 
. 6? (pd+ed?)— b? (pk +-ck* ) + 2ab./ (pd + cd? )sin. (—w)-+-+-a?sin.(— w) =) 2 
b? (p+ 2ck) Sia 
[d—k+cos.(—w)y]?...(16). These equations may be modified almost at pleasure, by assigning different values 
to the elements, and a vast variety of curves represented of singular forms and interesting properties, 
An equation embracing coefficients simpler than A, 6, &c., and in some 
cases more convenient in application, may be obtained-in a manner analo- 
gous to the preceding. Draw the tangent NA perpendicular to the axis of 
revolution CK, touching ARH at A. Let AN and the diameter ALM be’ 
the axes of the coordinates RL, AL, HM, AM. Complete the construc- 
tion as in fig. (1), and employ the same notation, as above. Since ARH is 
SNR VILE X “PA 
in (15), we derive from (4’) y? =| 
ve 
72 
a conic section, we have ne (pt+-ct?)=u*, in which (a’) is the diameter 
terminated at A, and (6’) its conjugate. Put t=>AM=AL+Qe=d+ 
