Sir W. Rowan Hamilton on Quatermofis. 133 



An equable spacing of the stars over the sky would seem to me 

 to be far more inconsistent with a total absence of Law or Prin- 

 ciple, than the existence of spaces of comparative condensa- 

 tion, including binary or more numerous groups, as well as of 

 regions of great paucity of stars. Thus, to take a familiar 

 instance : — No bad representation of stars and their distribu- 

 tion may be made by sparking viscid white paint from a coarse 

 brush upon a dark ground. It is impossible to conceive a 

 nearer approach to a "random scattering." But I am assured 

 by an ingenious friend, who has used this contrivance in aid 

 of pictorial effect, that such an artificial galaxy will present 

 every variety of grouping, with double and treble points in- 

 numerable (as I have indeed myself witnessed); nor can I 

 well see how upon any reasonable theory of chance it should 

 be otherwise. 



I wish to restrict this letter to the end proposed, that of 

 nakedly setting forth a serious difficulty in an inferential inter- 

 pretation of nature, sanctioned by high and also cumulative 

 authority. I shall not therefore attempt now to inquire more 

 minutely into the history of the error, if. error it be, nor to insist 

 on the great importance of arguing correctly in cases which 

 admit of so very extensive application. 

 I remain. Gentlemen, 



Yours faithfully, 



James D. P'orbes. 



XVIII. On Qtiaternions ; 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^ Sfc., Andrews' Professor of Astronomy in the 

 University of Dublin ^ mid Royal Astronomer of Ireland. 



[Continued from vol. xxxiv. p. 439.] 



82. ^I^HIS seems to be a proper place for inserting some 

 A notices of investigations and results, respecting the 

 inscription of rectilinear (but not generally plane) polygons, 

 in spheres, and other surfaces of the second degree. 



Let p and <r be any two unit- vectors, or directed radii of an 

 unit-sphere ; so that, according to a fundamental principle of 

 the present Calculus, we may write 



p2 = cr2=-l (237.) 



We sliall then have also, 



= <r"— p'^ = (r((r-p) + (<r-p)p, . . . (238.) 



