C 455 ] 
the equation *>*—t>‘Xm‘° J rm«l+n m x — dya‘*e 
y a 2, e — m c a e ■ — male — mn x e 
u = x -- and this being reduced, fhews, 
a 07 
that the fedfion P B is a line of the third order. 
9. If the diredtrix D N be a circle, and other things 
being as before, the fedtion will be a line of the 
fourth order. Fig. 9. 
Make the center of the circle to be in the line 
DL» parallel to A the ordinate to be N n — 
the abfciffe D n = z, the radius = r, and the equa- 
tion u 1 = ir z — z 2 . And let the plane paffing 
thro’ the vertical, cut the bafe in A R, and the fec- 
tion meet the diameter D K in E. Then the fame 
things being fuppofed, and the fymbols retained as 
before, we fhall have L n — z — / = z = 
n O OR = + M R = 
e a 
mac-\-ma l-\-mnx f 2r / a^-j- 2 r x n a — x^ r? — 2 a Inx — a r l * 
> V 
= u. And the analogy N O : P O ; : N R ; M R will 
. .. . mac-\-mal-}-mnx be — b x , 
give this equation, — — * — + 
y a* d~\-yqac-\-yqal-\-yqnx mac -\-mal-\-mnx~ a^y 
a 2 - a* 
v /trla x '\- 7 .rxna — x^ri 1 — 7. a Inx — a* l* . 
V a* 9 
which it appears, that the curve B P is a line of the 
fourth order. 
And in general it may be feen, that if the direc- 
trix of the bafe be a conic fedtion, except in the cafe 
above, the fedtion of the folid will be a line of the 
fourth order. 
lo> 
