PARALLELISM OF EDGES. 399 



equations of the projections upon one of the co- 

 ordinate planes, -of those intersections of the known 

 planes, to which the edges of the new plane are 

 respectively parallel. And from the necessary rela- 

 tion subsisting among the co-efficients of some of 

 the terms of these equations, the following equations 

 are derived. 



Let ,, <7 r , r be the unknown indices of the new 



.1 5 " i 5' 5 ' 



plane, which we chall call plane 5. Let /?,, </,, r,, 

 andp 2 , ^2, r 2 , be the known indices of two planes 

 to whose intersection one edge of plane 5 is parallel. 

 And let p 35 ^ 3 , r 3 , and p 4 , <? 4 , r 4 , be the known 

 indices of two other planes, to whose intersection 

 another edge of plane 5 is parallel. The particular- 

 values of the indices of the planes 1, 2, 3, and 4 

 being substituted for the general indices of those 

 planes in the following equations, the particular 

 values of the indices of plane 5 will be obtained. 



n v*i p 2 r 2 pj \q, r A r^qj 



(I) P* = 



i_ _ JL \ 

 2 p 4 p 3 yJ 





i \ / i i \ 



_ \fi ^ 2 ^ 2 ^ t 



r 5 / 1 1 



+f --LW x - ! 

 Hp^, ss! WTT: ^77; 



J L) f_L_-_L\ 



i <[* P 2 <ti' \!j'i ^ 4 ^4 ^ ro / 



