BOOK OF MAT] I KM ATI ( 'A K 1'UOBLEMS. 



1072. A triangle is projected orthogonally on each of tlnve 

 plant's mutually at right angles: prove that the algebraical sum 

 of the tetrahedrons which have these projections for bases and 

 a common vertex in the plane of the triangle is equal to the 

 tetrahedron which has the triangle for base and the intersection 

 of the plane for vertex. 



1073. A plane is drawn through the straight line 



x y z 

 7 = m = n : 



prove that the two other straight lines in which it meets the 

 surface 



(b c) yz (mz ny) + (c a)zx (nx Iz) + (a - b) xy (ly mx) = 

 are at right angles to each other. 



1074. If (?j, w,, TZ.J), (1 2 , m 2 , n 2 ), (1 3 , m 3 , n a ) be the direction 

 cosines of three straight lines which are, two and two, at right 

 angles, and if 



a b cab c 



then will 



a b c a 



- 4- + = : and =-r-. 



1075. The equations of the straight lines bisecting the angles 

 between the two straight lines given by the equations 



lx + my + nz = 0, ax a + by* + cz a = 0, 

 are 



Ix -f- my + nz = 0, lyz (b c) + mzx (c o)+ nxy (a b) = 0. 



1076. The straight lines bisecting the angles between the 

 two lines given by the equations 



Ix + my + nz = 0, ax 9 + "by 3 + cz* + 2a'yz + 2b'zx + Icxy = 0, 

 lie on the cone 



x*(c'n - b'm) +... + ... -f yz {cm ~ I' a + (c - b) 1} 4- ... + ... = 0. 



