102 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that P is the point 
(2 at + d \ 
(— ■ a ) 
the equation to PQ is 
d - 2 at( 2 at + d 
u 2a V 2 
and the equation to the locus of Q is 
4a 2 ?/ 2 = (d 2 - 4a 2 )x 2 - 2dx 3 . . . . (4) 
In particular, when d = zk 2a (4) becomes 
2/ 2 =+-* 3 (5) 
a 
which is Neil’s parabola. 
§ 14. In XVII the remark occurs : “ Curvas Omnes pure Hyperbolicas 
tertii Ordinis quae punctum duplex habent ad distantiam finitam de- 
scripsimus. Restant Gurvce Hyperbolo-Parabolicce et pure Parabolicee 
quarum Descriptiones facillimce ex methodo ipsius Prop. V deduci 
possunt .” 
[We proceed to discuss Maclaurin’s claim to have found a method for 
generating all rational cubics, by showing that his method is the simplest 
for obtaining the standard generation of these curves as given by 
L 0 + + tf 2 L 2 = 0 (1) 
M 0 + fM x =0 (2) 
Maclaurin proves in Part II that the pedal of a conic when the pole is 
at the focus is a circle for the central conic, and a straight line for the 
parabola. 
The converse also holds, and the analysis shows that the pencil of 
perpendiculars through the focus is in projective correspondence with the 
tangents to the conic, i.e. corresponding ray and tangent have equations 
of the form 
M 0 + tM 1 = 0 i 
Lq + £L X + = 0 f 
though not the most general of this kind. 
Let O x be the focus, and let 0 2 be another point the rays through which 
are in 1-1 correspondence with the rays through O x , so that corresponding 
rays intersect in R on a conic passing through O x and 0 2 . This latter conic 
may be replaced by a straight line without loss of generalisation. For let 
T be any point on it. Let 
L TOjC^ = a 
L T0 2 0 1 = /3 / . 
