Sir W. Rowan Hamilton on Quaternions. 34-3 



But also, by (131.), or by (132.), 



T>i = T.; T6=Tx; .... (136.) 

 and therefore, 



92_>j2 = x2-l2 (137.) 



Hence, by (135.), we obtain the expression, 



T(.-x)=^^; .... (138.) 



which may be substituted for the second member of the equa- 

 tion (133.), so as to complete the required elimination of i and 

 X. And if we then multiply on both sides by T()j— d), we 

 obtain this new form* of the equation of the ellipsoid: 



'r^^=*'-'" • • • • ('«»•) 



which will be found to include several interesting geometrical 

 significations. 



• This form was communicated to the Royal Irish Academy, at the stated 

 meeting of that body on March 16th, 1849, in a note addressed by the pre- 

 sent writer to the Rev. Charles Graves, It was remarlced, in that note, 

 that the directions of the two fixed vectors x, 6, are those of the two asym- 

 ptotes to the focal hyperbola; while their lengths are such that the two ex- 

 treme semiaxes of the ellipsoid may be expressed as follows : 



a=T)7+T(J; c=Tj7— T^j 

 the mean semiaxis being, at the same time, expressed (as in the text of 

 the present paper) by the formula 



i=T(„-0). 

 It was observed, further, that >j — 6 has the direction of o«<? cyclic normal of 

 the ellipsoid, and that jj-i — ^-i has the direction of the other cyclic normal; 

 that >j4-^ is the vector of one umbilic, and that >5-i + ^~' has the direction 

 o^ another umbilicar vector, or umbilicar semidiameter of the ellipsoid; that 

 t\\e focal ellipse is represented by the system of the two equations 



S.gU>j=S.eU^, 

 and 



TV.eU>5=2S Vn6, 



of which the first represents its plane, while the second, which (it was re- 

 marked) might also be thus written, 



TV.gU^rrgSV^, 

 re\ive?,ent% a. cylinder of revolution (or, under the latter form, a second cy- 

 linder of the same kind), whereon the focal ellipse is situated ; and that 

 the focal hyperbola is adequately expressed or represented by the single 

 equation, 



To which it may be added, that by changing the two fixed vectors n and 6 

 to others of the forms t-^m and t&, we pass to a confocal surface. 



[To be continued.] 



.,0 



