415 



And instead of treating (as has here been done) this third case, 

 or the case of contact at E, of the line 0' 0" with the circle (L), 

 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 A, B, we 

 might have treated it directly, by a shorter but less general 

 method.* 



13. The readers of the excellent Traite de Geometrie 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 nomographic 

 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 ijk, 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 formulas 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). 



2 s 



