26 ii:.l-Y*S.i.S' ACADEMY OF SCIEXCE. 



2. The central circles which intersect three fixed non-intersecting central 

 circles are generating circles of the same system of a pedal surface, and the three 

 fixed circles are generating circles of the opposite system of the same pedal sur- 

 face. 



1. Two straight lines, and two only, will, in general, meet each of four given 

 non-intersecting straight lines; but if the four given straight lines are all genera- 

 tors of the same system of a conict)id, then an infinite number of straight lines 

 will meet the four, which will all be generators of the opposite system of the same 

 conicoid. 



2. Two central circles, and two only, will, in general, meet each of four given 

 non-int3rsecting central circles; but if the four given circles are all generating 

 circles of the same system of a pedal surface, then an infinite number of central 

 circles will meet the four, which will all be generating circles of the opposite 

 system of the same pedal surface. 



1. Two planes are dra'wn, one through each of two intersecting generating 

 lines of a conicoid; these planes meet the surface in two other intersecting gen- 

 erating lines. 



2. Two central spheres are drawn, one through each of two intersecting gen- 

 erating circles of a pedal surface; these spheres meet the surface in two other 

 intersecting generating circles. 



1. The plane through the center of a conicoid and any generating line will cut 

 the surface in a parallel generating line, and will touch the asymptotic cone. 



2. The plane through the center of a pedal surface and any generating circle 

 will cut the surface in another generating circle equal to the first, and having a 

 common diameter with it, and will touch the tangent cone at the origin. 



1. The locus of the point of intersection of perpendicular generators of an 

 hyperboloid of one sheet is a sphere x- -\- y^ -\- z'^ = a'^ + ^' — C'. 



2. The locus of the point of intersection of generating circles of a pedal surface 

 of the "second kind" that cut orthogonally is a sphere given by the equation 



^' + 2/- + 2' = (a' -f &' — 0-2 f^ . 



This and the following theorem depend upon the fact that an angle in space 

 inverts into an equal angle. 



1. The angle between the generating lines through the point {x, y, z) of the 



x'- y'^ z^ fi\ + /?2 



quadric, — -\- — - -\- — = i, is cos -'^ , where /?!, /(a, are the roots of the 



a h c /h — /?2 



X- y^ z'^ 



equation, -I- -+- = 0. 



a{a — fi) b{b + /i) c (c + /?) 



2. The angle between the generating circles through the point {x, y, z) of the 



o;'^ if 2^ . fii — fh 



quartic, — -j- ■^~ + — == (^^ + 2/^ -\-^'^fj is cos —^ , where /ii, /is, are roots of 



a h c fix — /?2 



x'^ 7/2 ^2 



the equation + -I- = 0. 



a{a — K)^b(b + n)^G{G + ri) 



If a system of confocal conicoids be inverted from the center, a system of con- 

 Eocal pedal surfaces is obtained. The general equation of such a system is 



~^ + j:^ + t^=(-^' + 2/' + ^')' (30) 



a^ -\- n b- -\- n c^ -\- n 

 If fl is positive, the surface is a pedal of the "first kind," whose axes decrease 



