'144 Proceedings of the Royal Society of Edinburgh. 
If the line PP X is given by 
lx -|- my -1 = 0. 
;and OPj is _L r PP X through the origin 0, the co-ordinates of P x are 
i = l/ ( l 2 + m 2 ) ^ 
r] — m/(l 2 + m 2 ) ) 
Also 
l = iAP + r)Z) » 
m = rj/(£ 2 + rj 2 ) ) 
Hence if the equation in line co-ordinates to a curve is 
m) = 0 
its first positive pedal bas the equation in point co-ordinates 
.x 2 4 -y 2 x 2 + y 2 
= 0 
[Sess. 
• ( 1 ) 
• ( 2 ) 
• (3) 
• ( 4 ) 
• (5) 
Hence we may, in general, say that the degree of the first pedal is twice 
the class of the original curve. 
If 
/(*,</) = 0 ( 6 ) 
Is the equation in point co-ordinates to the first positive pedal of a curve, the 
curve itself has the equation in line co-ordinates 
l 111 
< 
l 2 + m 2 ' l 2 + m 2 
= 0 
• ( 7 ) 
.-and we should therefore say that the class of the latter is twice the degree 
«of its first pedal. 
The paradox arising from the application of these theorems is explained 
in the same way as for the degrees of a curve and its inverse, and both 
theorems are subject to important modifications. But the analysis given 
indicates the importance of line co-ordinates in the theory of pedals. 
Ex. A conic, being of class 2, has a pedal in general of degree 4, having 
;a special relation to the circular points at infinity. 
But if the pole O is at the focus the line equation to the conic is 
l 2 + m 2 + al + bm + c = 0 
.and its pedal is 
c(x 2 + y 2 ) + ax + by + 1 = 0, 
i.e. a circle. 
If the curve is a parabola 
al 2 -I- 2 him + bm 2 + 2 gl + 2/m = - 0 
the pedal is a circular cubic ; and if the focus is at O the pedal is the 
.straight line 
2 gx + 2fy + a - 0. 
