SOLID GEOMETRY. 231 



1100. Prove that the equation of the surface 



f.f *" ! 



a' + b'~? = l 



can be obtained in the form x* + y* z* = d 8 in an infinite number 

 of ways, provided that either a* or b* is greater than c 8 ; and that 

 the new axes of a;, y lie on the cone 



1101. The equation of a given hyperboloid may be obtained 

 in the form 



ayz + bzx + cxy = 1 



in an infinite number of ways; and, if a, /?, y be the angles be- 

 tween the axes in any such case, the expression 



abc 



1-cos'a - cos*/? - coa'y + '2 cos a cos ft cos y 

 will be constant 



1 1 02. Prove that the only conoid of the second degree i 

 hyperbolic paraboloid; and that it will be a right conoid, if the 

 principal sections be equal parabolas. 



1103. The equation 



aa?+ ... + ... -i- 1ayz+ ... 4- ... =0 

 will represent a cone of r.-volitti.. n, it' 



5V ^"-c'^cV | y t -a /> ^ 



a' l> c ~ ~F" r - a 



lint. The radios r of the central circular sections of the sur- 

 -bzx + cxy= 1 is given by the 



- 



-osinet of the sections by the equations 

 > 1 + n t ) m(n'--r) n 



