99 
The same construction may readily be inferred from 
Professor Lowery’s solution of a problem by Sir James 
Ivoiy, in Leybourn’s Mathematical Repository, vol. I, new 
series, page 175. 
Three solutions of this problem are given by Mr. Besant 
in his Geometrical Conics — art. 216, art. 217, and art. 249, 
Appendix. The first two of these solutions are not so 
simple as is the third, which would in all probability have 
been the same as Mr. Millar’s had he taken PE in his figure 
in the opposite direction. Mr. Besant derives his construc- 
tion from the investigation of the locus of a fixed point in a 
given straight line whose extremities move on the legs of a 
right-angled triangle. 
The various properties of the ellipse on which the solu- 
tion of this problem may depend are as follows, and they 
may be found demonstrated in most modern works on conic 
sections 
The lines AA' BB' are the principal diameters, CC' PP' 
are conjugate diameters, PF is perpendicular to OC, meeting 
the axes in G and g, PT is a tangent at P, and therefore paral- 
lel to CC'. Af A'T' are tangents at A and A!, 
PF , 
, PG = 
- OB- 
...(1) 
PF 
. Vg = 
0A= 
..(2) 
PG. 
OA ^ 
= OC . OB 
...(3) 
P^7 
. OB 
OC . OA . 
..(4) 
PG. 
p</ = 
- OCb.,... 
...(5) 
PF 
. 00 = 
OA.OB . 
..(6) 
PT . 
, Vt . 
= oc= 
...(7) 
OA 
+ 0B^ = 
OC^ + OP-' 
’■•(8) 
kt' . 
, AT' = 
:: OB" 
,.,(9) 
SP . 
S'P 
OC"- ....... 
(10) 
Vt' . 
PT' = 
= OC* 
.(11) 
