Sir W. Rowan Hamilton on Qtiatcrnions. 367 



root is essential. If we can effect this, we shall not only be able to 

 tell which roots are imaginary, but also to assign their values, if 

 such a term may be allowed in si^eaking of imaginaries. 



A very interesting discussion of the analogy between the methods 

 of Lagrange and Sturm, by Mr. Pv. Leslie Ellis, will be found in the 

 Cambridge Mathematical Journal, vol. ii. pp. 256-258. 



To the depression of equations and one or two other topics, 

 I do not think it desirable that these remarks should extend. 

 Apologizing to you for my prolixity, 



I remain, my dear Sir, 



Yours most truly, 



James Cockle. 

 To T. S. Davies, Esq., F.R.S. L. 8^- E., 

 Sj-c. Sfc. Sfc, iVoolxmch. 



XLIX. 071 Qiiaterniojis ; or on a New Si/sfem of Imaginaries 

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

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

 tute of France, SfC., Andrews' Professor of Astronomy in the 

 University of Dublin, and Royal Astronomer of Ireland. 

 [Continued from vol. xxxi. p. 519.] 

 5Q. TF we denote by b the length of the common radius of 

 J- the two diametral and circular sections, or the mean 

 semiaxis of the ellipsoid, which is also the radius of that con- 

 centric sphere of which the equation (24.) was assigned in 

 art. ^^j we shall have, by the formula (26.) of that article, the 

 following expression for this radius, or semiaxis: 



'='^ («'•> 



And hence, on account of the general formula, 



.p+pH=(,-H)(p+'^), . . . (82.) 



which holds good for any three vectors, i, k, p, the quaternion 

 equation of the ellipsoid may be changed from a form already 

 assigned, namely 



to the following equivalent form : 



T(p^-^-f±-^l)=:i (S3.) 



If then we introduce a new vector-.symbol A, denoting a line 

 of variable length, but one drawn in the fixed direction of 



