Sir W. Rowan Hamilton 07i Qiiaternions. 281 



process for the retraiidation of the geometrical construction * 

 of the ellipsoid described in article 31, into the language of 

 the calculus of quaternions, from which the construction itself 

 had been originally derived, in the manner stated in the 30th 

 article of this paper. Yet it may not seem obvious to readers 

 unfamiliar with this calculus, why the expression — p>tp~* was 

 taken, in that foregoing article 38, as one denoting, in length 

 and in direction, that radius o/the auxiliary sphere which was 

 drawn from c to d; nor in what sense, and for what reason, 

 this expression —p)cp~^ has been said to represent the reflexion 

 of the vector — X with respect to p. As a perfectly clear answer 

 to each of these questions, or a distinct justification of each of 

 the assumptions or assertions thus referred to, may not only 

 be useful in connection with the present mode of considering 

 the ellipsoid, but also may throw light on other applications 

 of quaternions to the treatment of geometrical and physical 

 problems, we shall not think it an irrelevant digression to enter 

 here into some details respecting this expression— pxp~^, and 

 respecting the ways in which it may present itself in calcula- 

 tions such as the foregoing. Let us therefore now denote by 

 0- the vector, whatever it may be, from c to d in the construc- 

 tion (c being still the centre of the sphere) ; and let us pro- 

 pose to find an expression for this sought vector a; as a func- 

 tion of p and of X, by the principles of the calculus of quater- 

 nions. 



40. For this purpose we have first the equation between 

 tensors, 



Tcr=Tx; (11.) 



which expresses that the two vectors o- and x are equally long, 

 as being both radii of one common auxiliary sphere, namely 

 those drawn from the centre c to the points d and a. And 

 secondly, we have the equation 



V.(o- — x)/5 = 0, (12.) 



where V is the characteristic of the operation of taking the 

 vector of a quaternion ; which equation expresses immediately 

 that the product of the two vectors cr — x and p is scalar, and 



* The brevity and novelty of this rule for constructing that important 

 surface may perhaps justify the reprinting it here. It was as follows : 

 From a fixed point a on the surface of a sphere, draw a variable chord au; 

 let d' be the second point of intersection of the spheric surface with the 

 secant bd, drawn to the variable extremity d of this chord ad from a fixed 

 external point b; take the radius vector ae equal in length to bd', and in 

 direction either coincident with, or opposite to, the chord ad; the locus 

 of the point e, thus constructed, will be an ellipsoid, which will pass through 

 the point b (and will have its centre at a). See Proceedings of the Royal 

 Irish Academy for July 1846. 



