[ 58 ] 



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

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

 V.P.Ll.LA., F.R.A.S., Corresponding Member of the Insti- 

 tute of France, tyc, Andrews' Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 



[Continued from voi.xxxii. p. 374.] 



62. rpHE equations (85.), (90.), and (111.), of articles 56, 

 J- 57, and 60, give 



T( P -K) = T( P -f,')=b', .... (113.) 

 and 



T( P -n) = T(p->,') = b; .... (114.) 



whence, by the meanings of the signs employed, the two follow- 

 ing mutually connected constructions may be derived, for geo- 

 metrically generating an ellipsoid from a rhombus of constant 

 perimeter, or for geometrically describing an arbitrary curve 

 on the surface of such an ellipsoid by the motion of a corner 

 of such a rhombus, which the writer supposes to be new. 



1st Generation. Let a rhombus lem'e', of which each side 

 preserves constantly a fixed length = b, but of which the angles 

 vary, move so that the two opposite corners l, m' traverse two 

 fixed and mutually intersecting straight lines ab, ab', (the point 

 l moving along the line ab, and the point m' along ab',) while 

 the diagonal lm', connecting these two opposite corners of the 

 rhombus, remains constantly parallel to a third fixed right 

 line ac (in the plane of the two former right lines) ; then, 

 according to whatever arbitrary law the plane of the rhombus 

 may turn, during this motion, its two remaining corners e, e' 

 will describe curves upon the surface of a fixed ellipsoid; which 

 surface is thus the locus of all the pairs of curves that can be 

 described by this first mode of generation. 



2nd Generation. Let now another rhombus, l'e"me'", with 

 the same constant perimeter =4 b, move so that its opposite 

 corners l', m traverse the same two fixed lines ab, ab', as before, 

 but in such a manner that the diagonal l'm, connecting these 

 two corners, remains parallel (not to the third fixed line ac, 

 but) to a fourth fixed line ac'; then, whatever may be the 

 arbitrary law according to which the plane of this new rhombus 

 turns, provided that the angles bab', cac', between the first 

 and second, and between the third and fourth fixed lines, have 

 one common bisector, the txvo remaining corners e", e'" of this 

 second rhombus will describe curves upon the surface of the same 

 fixed ellipsoid, as that determined by the former generation : 

 which surface is thus the locus of all the new pairs of curves, 

 described in this second mode, as it was just now seen to be 



