THE COLLECT 10X. HOOK Vili 
437 
so that PQ:QV = Q'W:PQ', 
or PQ.PQ'= QV.Q'W. 
Thus P can be found, and similarly P'. 
The conjugate diameter is found by virtue of the relation 
(conjugate diara.) 2 : PP' 2 = p: PP\ 
where p is the latus rectum to PP' determined by the property 
of the curve 
p: PP — AV 2 :PV.VP'. 
Problem, Given two conjugate diameters of an ellipse, 
to find the axes. 
Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are 
given two conjugate diameters, we can find the axes. The 
construction is as follows. Let AB, CP be conjugate diameters 
{CD being the greater), E the centre. 
Produce EA to H so that 
EA . AH = PE 1 . 
Through A draw FG parallel to CD. Bisect EH in K, and 
draw KL at right angles to EH meeting FG in L. 
With L as centre, and LE as radius, describe a circle cutting 
OF in G, F. 
Join EF, EG, and from A draw AM, AN parallel to EF, EG 
respectively.