290 SirW. Rowan Hamilton on Quaternions. 



or, substituting for the linear element dp the tangential vector t, 



S.vT«Tx = 0; (49.) 



or finally, by the principles of the same 20th article, 



VmK — XT<TV=0. (^O*) 



48. Under this last form, it was one of a few equations 

 selected in September 1846, for the purpose of being exhibited 

 to the Mathematical Section of the British Association at 

 Southampton ; although it happened* that the paper con- 

 taining those equations did not reach its destination in time to 

 be so exhibited. The equations here marked (49.) and (50.) 

 were however published before the close of the year in which 

 that meeting was held, as part of the abstract of a communi- 

 cation which had been made to the Royal Irish Academy in 

 the summer of that year. (See the Proceedings of the Academy 

 for July 184G, equations (46.) and (47.).) From the some- 

 what discursive character of the present series of communica- 

 tions on Quaternions, and from the desire which the author 

 feels to render them, to some extent, complete within them- 

 selves, or at least intelligible to those mathematical readers 

 of the Philosophical Magazine who may be disposed to favour 

 him with their attention, to the degree which the novelty of 

 the conceptions and method may require, without its being 

 necessary for such readers to refer to other publications of his 

 own, he is induced, and believes himself to be authorized, to 

 copy here a few other equations from that short and hitherto 

 unpublished Southampton paper, and to annex to them an- 

 other formula which may be found in the Proceedings, already 

 cited, of the Royal Irish Academy : together with a more ex- 

 tensive formula, which he believes to be new. 



49. Besides the equation of the ellipsoid, 



(/p + px)(p.-f?cp) = (x^-i2)2(21.), art.44; 

 with the expression derived from it, for the vector of proxi- 

 mity of that surface to its centre, 



(x2-,2)2y=(,2 + x2)^ + ,px-f xp. (31.), art. 45; 

 the equation for the lines of curvature on the ellipsoid, 

 VTiTx— xT<Tv=0 (50.), art. 47 ; 



and the equation VT-fTv = 0, (51.) 



which is a form of the relation S.vr = 0, that is of the equa- 

 tion (36.), article 45, of the present series of communications; 

 the author gave, in the paper which has been above referred 

 to, the following symbolic transformation, for the well-known 

 characteristic of operation. 



(d^) + \i\y) "^ (d J ' 



* See the note to article 45. 



