Sir W. Rowan Hamilton on Quaternions, 4t9,*J 



rules of this Calculus) for the two sliding spheres the two fol- 

 lowing more developed equations : 



Taking then the difference, and dividing by g, we find the 

 equation 



^(92->,2) = 2S.(fl-„)p; .... (148.) 



which, relatively to p, is linear, and may be considered as the 

 equation of the plane of the varying circle of intersection of 

 the two sliding spheres ; any one position of that plane being 

 distinguished from any other by the value of the coefficient g. 

 Eliminating therefore that coefficient g^ by substituting in 

 (146.) its value as given by (148.), we find that the equation 

 of the ellipsoid, regarded as the locus of the varying circle, 

 may be presented under either of the two following new forms : 



T(f-?%^0=*' • • ■ ("«•' 



t(p-?^^0=^=- • • C^"-) 



respecting which two forms it deserves to be noticed, that 

 either may be obtained from the other, by interchanging *j and 

 6. And we may verify that these two last equations of the 

 ellipsoid are consistent with each other, by observing that the 

 semisum of the two vectors under the sign T is perpendicular 

 to their semidifference (as it ought to be, in order to allow of 

 those two vectors themselves having any common length, such 

 as b) ; or that the condition of rectangularity, 



, (9 + »i)S.(a-~y,)p 

 P pIT^a -L9 — >Jj . • • (151.) 



is satisfied : which may be proved by showing (see Phil. Mag. 

 for July 1846) that the scalar of the product of these two last 

 vectors vanishes, as in fact it does, since the identity 



(9-,,)(9 + >,) = 6Hfl')->i9->}% 



resolves itself into the two following formulae: 



S.(d-„)(d+„)=fl^->,^1 



V.(d-,,)(fl + i,)=fl,-„d;J • • • V . •; 



of which the first is sufficient for our purpose. We may also 

 verify the recent equations (149.) (150.) of the eUipsoid, by 

 observing that they concur in giving the mean semiaxis b as 

 the length Tp of the radius of that diametral and circular sec- 



