Sir W. Rowan Hamilton on Quaternions* 425 



audition of the sounds emitted by separate points. Being 

 referred to an admitted phaenomenon, it is explained ; and 1 

 think I may say that it had not been completely explained 

 before. 



The conclusion of these researches is, therefore, that the 

 phaenomena of simultaneous perception of several sounds pro- 

 ceeding from the movement, whether of several points or of a 

 single one, are only modifications of one general phaenomenon, 

 which may be stated in the following manner : — 



" When our organ of hearing is affected by a movement 

 that may be geometrically decomposed into several others, 

 which, if they existed separately, would yield different sounds, 

 we generally perceive all these sounds at the same time." 



LXIII. On Quaternions; or on a New System of Imaginaries 

 in Algebra. By Sir William Rowan Hamilton, LL.D., 

 M.R.I. A. ^ F.R.A.S.i Corresponding Member of the Insti- 

 tute of France, ^c., Andrews* Professor of Astronomy in the 

 University of Dublin , and Royal Astronomer of Ireland. 

 [Continued from p. 343.] 



71. T>EFORE entering on any discussion of this new form 

 XJ of the equation of the ellipsoid, namely the form 



TV|j^j =fi'->l% eq. (139.), art. 70, 



it may be useful to point out another manner of arriving at 

 the same equation of the ellipsoid, by a different process of 

 calculation, from that construction or generation of the surface, 

 as the locus of the circle which is the mutual intersection of a 

 pair of equal spheres, sliding within two fixed cylinders of 

 revolution whose axes intersect each other; while the right 

 line, connecting the centres of the two sliding spheres, moves 

 parallel to itself, or remains constantly parallel to a fixed right 

 line in the plane of the fixed axes of the cylinders : which 

 mode of generating the ellipsoid was published in the Philo- 

 sophical Magazine for July 1848 (having also been communi- 

 cated to the Royal Irish Academy in the preceding May), 

 as a deduction from the Calculus of Quaternions. And 

 whereas the fixed right line, through the centre of the ellip- 

 soid, to which the line connecting the centres of the two sli- 

 ding spheres is parallel, may have either of two positions, 

 since it may coincide with either of the two cyclic normals, 

 we shall here suppose it to have the direction of the cyclic 

 normal », or shall consider the second pair of sliding spheres 



