S84 Sir W. Rowan Hamilton 07i Quaternions, 



in these calculations, to indicate indeed the lengths atid direc- 

 tions, but not the situatio?iSf of the straight lines which they 

 are employed to denote. 



43. The same geometrical interpretation of the symbol 

 —pxp~^ may be obtained in several other ways, among which 

 we shall specify the following. Whatever the lengths and 

 directions of the two straight lines denoted hyp and x may be, 

 we may always conceive that the latter line, regarded as a 

 vector, is or may be decomposed, by two different projections, 

 into two partial or component vectors, x' and x", of which one 

 is parallel and the other is perpendicular to p ; so that they 

 satisfy respectively the equations of parallelism and perpen- 

 dicularity (see article 21), and that we have consequently, 



x = x'-fx"; Y.x'p=0; S.x"p = 0; . . (18.) 



where S is the characteristic of the operation of taking the 

 scalar of a quaternion. The equation of parallelism gives 

 px.' = x'p, and the equation of perpendicularity gives px"= —x"p ; 

 hence the proposed expression --pxp-^ resolves itself into the 

 two parts, 



-px'p-^=:-x'/5p-»=-x'; \ ^^. 



^pK"p-'=+x"pp-'=-}-x";J • • • ^ •-' 

 so that we have, upon the whole, 



-pXp-'=-p{K' + K")p-'-->J-^K". . . (20.) 



The part — x' of this last expression, which is parallel to p, is 

 the same as the corresponding part of — x; but the part +t", 

 perpendicular to p, is the same with the corresponding part 

 of + X, or is opposite to the corresponding part of — x ; we 

 may therefore be led by this process also to regard the expres- 

 sion (17.) as denoting the reflexion of the vector — x, with 

 respect to the vector p, legarded as a reflecting line ; and we 

 see that the direction of p, or that of —p, is exactly interme- 

 diate between the two directions of — x and —pxp~^f or be- 

 tween those of X and of pxp~'. 



4-4-. The equation (9.) of the ellipsoid, in article 38, or the 

 equation (4.) in article 37, may be more fully written thus: 



(,p+px)(p. + xp) = (x2-.T. . . . . (21.) 



And to express that we propose to cut this surface by any 

 diametral plane, we may write the equation 



OTp+pOT=0, (22.) 



where ■bt denotes a vector to which that cutting plane is 

 perpendicular: thus, if in particular, we change ot to x, we 

 find, for the corresponding plane through the centre, the equa- 

 tion 



