Sir W. Rowan Hamilton on Qiiaternio7is. 373 



1847; or the Proceedings of the Royal Irish Academy for 

 July, 1846.) Hence the two new sides b'c' and c'a, which 

 indeed are parallel by (107.) to the two old sides AC and cb, 

 or to X and i, must have the directions of the two cyclic nor- 

 mals ; and the third new side, ab', or *' — x', must be the axis 

 of a second cijli7ider of revolution, circumscribed round the same 

 ellipsoid. If we determine on this new axis two new points, 

 l' and im', as the extremities of two new vectors a' and ^', 

 analogous to the recently considered vectors A and ^, and 

 assigned by equations similar to (84.) and (89.), namely 



a'(x'-,') = hV + p', p'(.'-x') = .V + p.', . . (110.) 

 we shall have results analogous to (85.) and (90.), namely 



T(p-A') = 6; T(p-fx') = ^5 • • • (111-) 

 with others similar to (101.), namely 



0-A' ^-f.'_A'-,.\ 



,' ~" ,' ~ i' — v/ ' 



(112.) 



the common value of these three quotients being a new scalar, 

 but f being still the same vector as before, namely that vector 

 which terminates in the point n, where the normal to the sur- 

 face at E meets the common plane of the new and old gene- 

 rating triangles, or the plane of the greatest and least axes of 

 the ellipsoid. It is easy hence to infer that the new variable tri- 

 angle l'm'n is similar to the new generating triangle ab'c', and 

 similarly situated in the same fixed plane therewith ; and that 

 the sidesL'N,M'N, having respectively the same directions as ac', 

 b'c', have likewise the same directions as Bc, ac, and therefore 

 also as MN, LN, or else directions opposite to these ; in such a 

 manner that the two straight lines, l'm, m'l, must cross each 

 other in the point N. But these two lines may be regarded 

 as the diagonals of a certain quadrilateral inscribed in a circle, 

 namely the plane quadrilateral l'm'ml ; of which the four cor- 

 ners are, by (85.), (90.), and (111.), at one common and con- 

 stant distance =b, from the variable point E of the ellipsoid. 

 If then we assume it as known that the vector U^v, which is in 

 direction opposite and is in length reciprocal to the perpen- 

 dicular let fall from the centre a on the tangent plane at e, 

 must terminate in a point f on the surface o{ another ellipsoid, 

 reciprocal (in a well-known sense) to that former ellipsoid 

 which contains the point e itself, or the termination of the 

 vector p ; we may combine the recent results, so as to obtain 

 the following geometrical construction*, which ?,ex\esto gene- 



* This is the construction rerci-re.l to in a note to article 54. It was 

 comiiuinicatcil by the aiiihoi- to tlie Royal Irish Acailcmy, at the meeting 

 of November M, l!i47. Sec the Proceedings of that date. 



