280 Sir W. Rowan Hamilton on Quaternions. 



variable point e of the surface ; so that the constant vector 

 I — X is, by the construction, a particular value of this variable 

 vector p. The vector from a to c, beiufr the opposite of that 

 from c to A, is denoted by — x; and if d be still the same 

 auxiliary point on the surface of the auxiliary sphere, which 

 was denoted by the same letter in the account already printed 

 of the construction, then the vector from c to d, which may 

 be regarded as being (in a sense to be hereafter more fully 

 considered) the reflexion of — x with i-espect to ^, is = —pxp-'^; 

 and consequently the vector from d to b is =j-j-pxp~^ The 

 lengths of the two straight lines bd, and ae, are therefore re- 

 spectively denoted by the two tensors, T(i + ^x^-') and Tp; 

 and the rectangle under those two lines is represented by the 

 product of these two tensors, that is by the tensor of the 

 product, or by T(»p4-px). But by the fundamental equality 

 of the lengths of the diagonals, ae, bd', of the plane quadri- 

 lateral ajbed' in the construction, this rectangle under bd and 

 ae is equal to the constant rectangle under bd and bd', that 

 is under the whole secant and its external part, or to the 

 square on the tangent from b, if the point b be supposed ex- 

 ternal to the auxiliary sphere, which has its centre at c, and 

 passes through d, d', and a. Thus T{ip + pK) is equal to 

 (Ti)^— (Tx)% or to x.^ — 1% which difference is here a positive 

 scalar, because it is supposed that cb is longer than ca, or that 



Ti>Tx; (8.) 



and the quaternion equation (6.) of the ellipsoid reproduces 

 itself, as a result of the geometrical construction, under the 

 slightly simplified form* 



T(i^ + px) = x2-.2 (9.) 



And to verify that this equation relative to p is satisfied (as 

 we have seen that it ought to be) by the particular value 



P = '-x, (10.) 



which corresponds to the particular position b of the variable 

 point E on the surface of the ellipsoid, we have only to observe 

 that, identically, 



<(» — 1) + (< — x)x = »^ — »X + JX— x^ 



and that (by article 19) the tensor of a negative scalar is equal 

 to the positive opposite thereof. 



39. The foregoing article contains a sufficiently simple 



* See the Proceedings of the Royal Irish Academy for July 1846, equa- 

 tion (44.). 



