Sir W. Rowan Hamilton on Qjiaternions, 215 



soid, by a suitable choice of the six real constants ImnV rri 11!, 

 At the same time the equation 



will represent a system of two parallel planes, which touch 

 the ellipsoid at the extremities of the diameter denoted by the 

 equation 



v = 0', 



and this diameter will be the axis of revolution of a certain 

 circumscribed cylinder, namely of the cylinder denoted by 

 the equation 



the equation of the plane of the ellipse of contact, along which 

 this circular cylinder envelopes the ellipsoid, being, in the 

 same notation, 



M=0: 



all which may be inferred from ordinary principles, and agrees 

 with what was remarked in the 29th article of this paper. 



34. This being premised, let us next introduce three new- 

 constants, p, g, r, depending on the six former constants by 

 the three relations 



2p = l + l'j 2q = m + m'f 2r=zti-\-n'. 



We shall then have 



I'x + m'j/ + n'z = 2 {px -\-qi^-\-rz)— [Ix -\-my-^ nz) ; 



and the equation (1.) of the ellipsoid will become 



{W -\-mm' -^nn'f 

 - {p + m^ + n^)(^x^ ^f + 2'^) 



—^{lx-\- my + nz)[px -\-qy-{-rz) 

 + ^{px-^qy-\-rzY 



if we introduce three new variables, of, y\ s', depending on 

 the three old variables .r, y^ z, or rather on their ratios, and 

 on the three new constants p, q, r, by the conditions, 



*' _ V _ ^' _ 2{px-{-qy + rz) 

 x~ y z x^-\-y^-\-z^ 



These three last equations give, by elimination of the two 

 ratios of ,r, y, z, the relation 



^'2 ^ya ^ ^n = 2(/?y + qy' + rz') ; 



the new variables a/, y\ z' are therefore co-ordinates of a new 

 point, which has for its locus a certain spheric surface, passing 

 through the centre of the ellipsoid ; and the same new point 



