436 
PROFESSOR J. CASEY ON A NEW 
Cor. If the point O be within the circle J, we shall have the sum of the diameters 
equal the diameter of J. 
139, If we denote the diameter of J by 2g, and if a line OQQ' (see last diagram) 
p/Q/ PQ, 
infinitely near OP make an angle dO with OP, then -^-=P'M and -^-=PL. 
Hence by art. 138 we have 
P'Q'-PQ=2 ? ^. 
(292) 
Cor. If the point O be inside J we have 
P'Q'+PQ=2 ? <7fl (293) 
140. If four circles be mutually orthogonal, and if any figure be inverted with respect 
to each of the four circles in succession, the fourth inversion will coincide with the 
original figure *. 
Demonstration. — It will plainly be sufficient to prove 
the proposition for a single point, for the general proposi- 
tion will then follow. 
Since the four circles are mutually orthogonal, their 
four centres will form the angular points and the inter- 
section of the perpendiculars of a plane triangle. Let 
them be the points A, B, C, O ; CO produced will inter- 
sect AB perpendicularly in D, and the squares of the 
radii of the four circles will be equal to the four rect- 
angles 
AB . AD, BA.BD, -CO . OD, CD . OD, 
one of the circles being imaginary, namely the one the square of whose radius is 
— CO . OD. Now let P be the point we operate on, and let P' be its inverse with 
respect to the circle A, and P" the inverse of P' with respect to the circle B. Join P"0 
and CP meeting in P'". Now since P' is the inverse of P with respect to the circle A, 
the square of whose radius is AB . AD, we have the rectangle AB . AD=AP . AP'. 
The triangle ADP is similar to the triangle AP'B, therefore the angle ADP=angle AP'B ; 
in like manner, the angle BDP"= angle AP'B, therefore the triangles ADP and BDP" 
are equiangular, and the rectangle AD . DB=rectangle PD . DP". Again, because O 
is the intersection of the perpendiculars of the triangle ABC, the rectangle AD . DB= 
CD . OD ; hence CD . OD=PD . DP", and the angles CDP and P"DO being the com- 
Fig. 23. 
* An important extension of this theorem can he got hy combining it with the following proposition, which 
is proved in art. 95 of my memoir on “ Cyclides and Sphero-Quartics ” : — If a sphero-quartic he projected on 
one of the planes of circular section of any quadric passing through it hy lines parallel to the greatest or least 
axis of the quadric, the projection will he a hicircular quartic whose centres of inversion will he the projection 
of the centres of inversion of the sphero-quartic. The extension is as follows. There exists in sphero-qucirtics 
a series of inscribed quadrilaterals ABCD, whose sides AB, BC, CD, DA, taken in order, pass through the vertices 
of the four cones of the sphero-quartic. 
