A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

CHAPTER VII.
INVERSIONS.
SECTION I.-INVERSION IN SPACE.
113. DEF. XXXVII.-Being given a fixed point S and a
system ofpoints A, B, C..... if upon the right lines SA, SB,
SC a system of points A', B', C'.... be determined by the relation SA. SA' = SB. SB' = SC. S C,.c. = constant, say k2, the
two systems A, B, C... A'B'C'.... are said to be inverse of
each other. The point S is called the centre of inversion, and the
sphere whose centre is S and radius k, the sphere of inversion.
114. The figure inverse to a plane is a sphere passing through
the centre of inversion.
S
A5XB
A B
Fig. 45.
DEM.-From S draw the right line SA perpendicular to the
plane P, and in P draw any line AB through A; then (Sequel,
Prop. xx., p. 41), the inverse of the line AB is a circle SA'B'
passing through S. Now, if the whole figure, consisting of the