as eee 
_— 
ae 
a) BSE 
= 
. 
a 
' 
a 
a 
% 
4 
r 
415 
And instead of treating (as has here been done) this third case, 
or the case of contact at E, of the line O’O” with the circle (LZ), 
as a limit of the first case, or of the case of intersection of that 
line with that circle in two real and distinct points 4, B, we 
might have treated it directly, by a shorter but less general 
method.* 
13. The readers of the excellent Traité de Géometrie de 
Position, by M. Chasles (Paris, 1852), with which the author 
of the present paper does not pretend to be more than partially 
acquainted, will not fail to recognise the double homographic 
division of the radical axis (whence such divisions on the cir- 
cular loci can easily be obtained), with the double points A, B, 
and with O’, O” as homologues of infinity. That theory of 
homographic division may also be employed in the treatment 
of the case where A and B become imaginary, without any 
previous reference to the case where those two points are real. 
It was, however, almost entirely through the quaternion me- 
thod, including, indeed (as lately stated to the Academy), some 
use of biquaternions, or combining the employment of the old 
imaginary of algebra with that of his peculiar symbols 7, that 
Sir W. R. H. was led, not merely to the results, but even to 
the chief constructions of the present paper. In particular, he 
was led to perceive the theorem of circulation in Art. 11, and 
to make out the geometrical construction given in that article 
for exhibiting it, by endeavouring to interpret formule which 
presented themselves to him, in investigating the integral of 
an equation in finite differences of quaternions, which integral 
was found to contain a periodical term. 
* Some remarks on this case have appeared in the number of the Philoso- 
phical Magazine for the present month (April, 1853). 
VOL. V. 2s 
