';.} (70.) 



Sir W. Rowan Hamilton on Qiiaiernions. 517 



then, on substituting these values for i and x in the condition 

 (4-9.), or in the equation 0=S.vt<tx, the terms involving i" 

 and x" will vanish of themselves, and the equation to be satis- 

 fied will become 



0=S. vtj'tx'; (69.) 



which is thus far a simplified form of the equation (49.), that 

 three of the four directions to be compared (namely tho!>e of 

 *', x', and t) are now parallel to one common plane, namely to 

 the plane which touches the ellipsoid at the proposed point, 

 and to which the fourth direction (that of v) is perpendicular. 

 Decomposing the two quaternion products, t*' and tx', into 

 their respective scalar and vector parts, by the general formulae, 



Tl'=S.Tl' + V.T*'; 

 TX'=S.TX' + V.TX' 



and observing that the vectors V. n' and V. tx' both represent 

 lines parallel to v, because v is perpendicular to the common 

 plane of t, »', x'; so that the three following binary pi'oducts, 

 V.Tj'.V. tx', vV. t«', vV. tx', are in the present question scalars ; 

 we find that we may write 



S.vt«'tx' = vS.t/.V.tx' + vV.t.'.S.tx'. . . (71.) 



Hence the equation (69.) or (49.) reduces itself, after being 

 multiplied by v~', to the form 



S.t/.V.tx' + V.t.'.S.tx'=0; .... (72.) 



which gives, in general, by the rules of the present calculus, 



V.t'T _ V.Tx' 



sT?7-"sr;v' ^^^'> 



and by another transformation, 



V.i't-1 _ V.x't-' , ^ 



[s:?F^-~s:^?^' (^^•) 



which may perhaps be not inconveniently written also thus : 

 V i' _ V x' 

 S"'f"'"S*7' C^^') 



in using which abridged notation, we must be careful to re- 



V 



member, respecting the characteristic ^, of which the effect 



is to form or to denote the quotient of the vector 'part divided 

 by the scalar part of any quaternion expression to which it is 

 prefixed, that this Jiew characteristic of operation is not (like S 

 and V themselves) distributive relatively to the operand. The 

 vector denoted by the first member of (74.) or of (75.) is a line 

 perpendicular to the plane of *' and t, that is to the tangent 



