SOLID GEOMETRY. 229 



1090. By transformation of co-ordinates, prove that the equa- 

 tion 



x* + y* + z* + yz + zx + xy = a* 



-ents an oblate spheroid whose polar axis is to its equatoreal 

 in the ratio 1:2; and the equations of whose polar axis arc 

 x = y = z. 



1091. If a cone of the second degree touch one system of 

 planes, which are two and two at right angles, it will touch 



an infinite number of such systems: and if one system be co-ordi- 

 nate planes, and (/,, m |t n,), (J a , m a , w a ), (J 3 , w 3 , n z ) be the direc- 

 tion cosines of another; the equation of the cone will be 



1092. Prove that the surface whose equation, referred to 

 axes inclined each to each at an angle of GO , is 



yz + zx + xy + a 9 = 0, 

 is cut by the plane x + y + z = in a circle whose radius is a. 



1093. In the expression 



ox* + 6y* + cz* + 2ayz + 2b'zx + 2c'xy + 2a"x + 11" y + 2c"z + d; 



prove that 



(6 4- c) a" 9 + . . . -f . . . - a'b "c" - b'c"a" - c'aT', 



and a'"(6c-O+ ... + ... 4- 2&V'(&V-aa') + ... + ... 



are invariants for all systems of rectangular co-ordinates having 

 the same origin. 



1094. Prove also that the coefficients in the following equa- 

 tion i 



A + o, AcosyW, hco&P + b', a" =0, 



Aooay + c', H + b, A cos a + a', 6" 



A cos/? + 6', h cos a + a', h + <?, c" 



i", c", c* 



& 7 Ix-ing the angles between the axes, are invariants for all 

 systems of co-ordinates having the same origin. 



