SOLID GEOMETRY. 



1112. Prove that the locus of tangent lines, drawn from the 

 .11 to the surface 



u = cu? + . . . 4- 2a'ys 4- . . . + 2a"x + ... +/= 0, 

 is /M - (a"x + 6"y + c"s +/)' = ; 



:uul investigate the condition that the surface may be a cone from 

 consideration that this locus will then become two planes. 



1113. The section of the surface yz + zx + xy = a 9 by the 

 J x + my + nz = p will be a parabola if 



and that of the surface 



x* + y* + - 2yz- 2zx- 2xy = a* 



will be a parabola if ^ 



mn + nl + Im = 0. 



1114. The semiaxes of a central section of the surface 

 ayz + bzx + cxy + abc = 



jii.i.l. 1 y :i plane whose direction-cosines are I, m, n, are given by 



the equation 



r 4 (2lcmn + ... - oV -...)- iaicr 1 (aw n +...) + 4a f 6V = 0. 



. Prove that the section of the surface 



tut +... + 2a'yz+ ... + '2a"x + ... +<J=0 

 l.y tin- i-lane ^x-t-wy-f- 7t5 = will be a rectangular hypi-rl..-; 

 r(6-f-c)4 m^c-f-oj + n^a + ft)^ - 

 a parabola, if 



r(6c-a /f )-f-... -f. .//c'-aoV ... + ... =0. 



i his last equation becomes identical if 

 6V=oa', cV = 66', and a'6' = cc. 



