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AND OTHER SYSTEMS OF TWO TETRADS. 
427 
5. The blue circles have a common orthotomic circle Q, that is, the radical 
axis or common chord of each two of the blue circles passes through one and the 
same point, the centre of the circle i2. The figure is symmetrical in regard to the 
red and blue circles respectively, and thus the red circles have a common orthotomic 
circle O', that is, the radical axis or common chord of each two of the red circles 
passes through one and the same point, the centre of the circle il'. 
6. Projecting stereographically on a spherical surface, the four red circles and 
the four blue circles become circles of the sphere; and then making the general 
homographic transformation, they become plane sections of a quadric surface; we have 
thus the theorem that on a given quadric surface it is possible to find four red 
sections and four blue sections such that each blue section touches each red section; 
and moreover the capacity of the system is = 9; viz. 3 of the red sections may be 
assumed at pleasure. But (as is well known) the theory of the tangency of plane 
sections of a quadric surface is far more simple than that of the tangency of circles: 
the condition in order that two sections may touch each other is simply the condition 
that the line of intersection of the two planes shall touch the quadric surface. And 
we construct, as follows, the sections touching each of three given sections: say the 
given sections are 1, 2, 3; through the sections 1 and 2 we have two quadric cones 
having for vertices say the points d 12 and i 12 (direct and inverse centres of the two 
sections): similarly, through the sections 1 and 3 we have two quadric cones vertices 
d 13 and i 13 respectively, and through the sections 2 and 3 we have two quadric cones 
vertices d 23 and i 23 respectively; the points d 12 , i 12 , d 13) i 13 , d 23 , lie three and three 
in four intersecting lines or axes, viz. these are d^d^d^, d 23 i 31 i 12 , d 31 i 12 i 23 , d 12 i 23 i 31 
respectively. Through any one of these axes, say d 23 d sl d 12 , we may draw to the 
quadric surface two tangent planes each touching the three cones which have their 
vertices in the points d 23 , d 31 , d 12 respectively; and the section by either tangent 
plane is thus a section touching each of the three given sections 1, 2, 3; we have 
thus the eight tangent sections of these three sections. 
7. Taking as three of the red sections the arbitrary sections 1, 2, 3; and 
grouping together two at pleasure of these sections, say 2 and 3; we may take for 
the blue sections the two sections through the axis d 2S d 31 d 12 , and those through the 
axis d^i^i^-, we have thus the four blue sections touching each of the given red 
sections 1, 2, 3; and this being so, there exists a remaining red section 4 touching 
each of the blue sections; we have thus the four blue sections touching each of the 
red sections 1, 2, 3 and 4. This implies that the vertices or points d 2i and d 3i lie 
on the axis d 23 d 31 d 12 , and that the vertices or points f 24 and lie on the axis 
d-xhihi; or, what is the same thing, that the four sections 1, 2, 3, 4 have in 
common an axis d. a d 2l d. 3l d^d^ and also an axis 
8. If the quadric surface be a flat surface (surface aplatie) or conic, then the 
red sections become chords of the conic; the axes are lines in the plane of the 
conic, and thus the tangent planes through an axis each coincide with the plane of 
the conic, and it would at first sight appear that any theorem as to tangency 
becomes nugatory. But this is not so; comparing with the last preceding paragraph, 
we still have the theorem: on a given conic, taking at pleasure any three chords 
54—2