100 
The properties 7, 8, 9, and 10 are true for any conjugate 
AA' and BB'. 
Emerson, in his Conic Sections, published in 17G7, solved 
this problem by means of property 7. Sir J. Leshe, in his 
Geometry of Curved Lines, pages 255 , 256 ; and Salmon, in 
his Conic Sections, art. 179, page 168, fifth edition, have 
repeated Emerson’s solution. 
The radius of the generating circle used by Mr. Millar is 
determined in terms of the principal axes by Wallace in his 
Conics somewhat in the same manner as that used by Mr. 
Millar. 
This problem is given as an exercise by C. Taylor, in his 
Conics, published at Cambridge in 1863, ex. 9, page 90. 
Several eminent writers do not refer to it. Amongst this 
number are Todhunter, Hymer, Drew, Hamilton, Hustlei’, 
Bridge, Buckle, and Jackson. 
In the Oxford, Cambridge, and Dublin Messenger of 
Mathematics, vol. III. page 151, R. Tucker, M.A., has given 
a solution of this problem. The construction of the mag- 
nitude of the axes is simple; their directions, however, are 
not so simple, as they are made to depend upon the focal 
properties of the ellipse. 
This provoked another solution in the same volume, page 
227, by R A. Proctor, B.A., whose construction is very 
simple indeed. The lines employed by Mr. Proctor are dif- 
ferent from those employed by Mr. Millar, but the construc- 
tion of each is about equally simple. 
In all the cases above referred to the construction of the 
principal axes only is given. The direct construction of any 
pair of conjugate diameters from the first pair is not con- 
sidered by any of the above named authors. The following 
is offered as a solution of this part of the problem. I am 
not sure it is the simplest that could be devised. It depends 
upon property (7) when the axes are not the principal 
axes. 
