[383 
383] 
PROBLEMS AND SOLUTIONS. 
605 
y) 
nstruct, 
b that 
hat an 
8, and 
square, 
tieorem 
erefore 
I. The 
n : the 
rishes.} 
of the 
meters 
Solution by Professor Cayley. 
Let the given line be taken for the axis of z; the axes of x, y being any 
rectangular axes in the plane perpendicular thereto ; the equations of the given line 
are therefore (« = 0, y = 0). Take (a,, ft, Yi), (a,, ft, y 9 ), («*, ft, y 3 ), («4, ft, 74) for the 
coordinates of the points A, B, G, D respectively; and (0, 0, eft (0, 0, c 2 ), (0, 0, eft (0, 0, c 4 ) 
for the coordinates of the points a, b, c, d respectively. Then, to determine Cj, the 
equation of the plane BCD is 
«, y , 
•2 , 
1 
a.,, ft, 
72, 
1 
«3, ft, 
73, 
1 
®4 , ft, 
74, 
1 
and cutting this by the line x = 0, y = 0, we have 
^ 0 , 0 , Ci, 
«2, ft, 72, 
«8» ft, 73, 
! a 4 , ft, 74, 
with similar equations for c 2 , c 3 , c 4 respectively. The four equations may be united into 
the single equation 
CiPit 
1, 
«1, 
ft 
= 
pi, 
«1, 
ft, 
7i 1 
c,p,, 
1, 
«2, 
/3. 
P-2, 
«2, 
ft, 
72 ; 
\ C 3 p 3 , 
! 
1, 
«3, 
ft 
p3, 
«3, 
ft, 
73 
Cipi, 
1, 
«4, 
ft 
Pi, 
«4, 
ft, 
74 
where p 1} p 2 , p 3 , pi are arbitrary multipliers. Hence, writing successively (p 1 , p. 2 , p 3 , p 4 ) 
= (1, 1, 1, 1) and (pi, p 2 , p 3 , pi) = ( 7 i, 72, 73, 74), we have first 
that is 
^1) 
1, 
«1, 
ft 
= 
1, 
«1, 
ft, 
7i 
C*> 
1, 
«8, 
ft 
1, 
«2, 
ft, 
72 
C.3) 
1, 
«8, 
/ft 
1, 
«3, 
ft, 
73 
C4) 
1, 
«4, 
ft 
1, 
Ä4, 
ft, 
74 
1, 
«1, 
ft, 
Cl 
+ 7i 
= 0 
1, 
«2,- 
ft 
c 2 
+ 7.4 
1, 
«3, 
ft 
C3 
+ 73 
1, 
*4, 
ft, 
C4 
+ 74