230 PROFESSOE C. J. JOEY OX QUATEEXIOXS AXD PEOJECTIYE GEOMETRY. 
8. The united-points of a function cmd of its conjugate form reciprocal tetraliedra 
u'ith respect to the unit sgyhere = 0. 
For when the roots are all unequal 
if q\ q'. 2 , q's ^' 4 . the united j^oints of the conjugate. Thus the jDoiuts q^ and 
q '2 are conjugate with respect to the sphere. 
Since the plane Sq\q = 0 contains tire points cq^, q^, q^, the weights may he chosen 
so that 
iMsgJ A ^ \MMA n' = (oq\ . 
and these relations imply"^ 
S^i^'i = Sg.3(7'3 = = ^qgqj = 1 
and from symmetry 
(29); 
a — [ 7 g*/3*i' +] . _ _[7 VzVZji1_ ,, — [747 7/'gl _ r7'i7V/U /..a\ 
(7'i7W*)’ {7'.7'87'47 'i)’ * (7'37Vi'i7A ~ (yViViVi's)' ' '' 
To these relations may be added the quaternion identities 
+ 222^3 + • . . . (31), 
qfqj + qf>9.'-i + Al'z + 24 ^ 2^4 = 1 = q'fqi + qf>T 2 + + 2 ^ 4^24 • ( 3 -), 
udiich are probably more elegant than important. The second shows that the centre 
of the sphere is the centre of mass of the weights Sq^Sq^, Sp^Sf/o, SpgS^'g, Sf/^^S^'^ 
placed at the vertices of either of the tetrahedra, and that the sum of their weights 
is unity. 
From these identities we deduce the vector equations 
( 2 i 2 ^) + (222^2) + (232^3) + (2424) — 0 — + [222^2] + [232^3] + [242 4] ( 33 ), 
which express that equilibrating forces can be placed along the hnes joining 
corresponding vertices, or that any line which meets three of these lines meets the 
fourth, or that the lines are generators of a cpiadric.t 
* Writing ^i = Wi (1 + a.i), (1 + a'i), equations (29) give Wiw'i (1+Saia'i) = 1. Hence the 
jiroduct of the weights v:iw\ is the reciprocal of the product of the perpendiculars from the centre of the 
sjjhere and from the point qi (or 2i) on the opposite face of the tetrahedron q\qdio,qi (or q\(liiz'ld- 
Observe that only the jiroducts wiw\ have l^een assigned, not v:\ and ta'i separately. 
t In the notation of the last note (33) becomes Et/yqw'i (a'l — ap = Xit'iw'iYaia'i = 0. The equilibrating 
forces are proportional to the distances between the vertices divided by the products of perpendiculars 
mentioned in the note cited. 
