128 



Journal oe the Mitchell Society. 



[Dec. 



In like manner, since the plane c may be written in any one 



of the forms 



17 (32 x + 27 w) — 64 (4 x + 3y + 12 z) = 0, 

 32 (3 x + 4 y — 8 *) — 3 (64 j — 51 w) = 0, 

 32 (3 x — 2y + z) — 9 (32 z — 17 w) = 0, 



the equations of lines cf, eg, and c/j are as follows: 





 



\2>x + 4y — Sz=Q 

 (64y — 21iv = 

 x — 2y -r z = 

 z — 17 w =0. 



cf: 

 eg: 

 eh: 



{4x + 3y + 12z 

 I 32 x + 27zu 



( 2>x 

 (32 



We have now to determine the equations of the lines 1, 2, 

 3, 4, 5, 6, each one of which passes through the point (0, 0, 

 0, 1) and is conditioned as stated in § 1. Considering line 1, 

 its equations must be of the form 



x + A y + ixz = 

 x -f Ky + /*,.*= 



and since it meets line of, we have the condition 



1, 0, 0, =0 



0, 0, 0, 1 



1, X, p, 

 1, DA,, /*,, 



giving A : A t = /u. : fx z and hence line 1 may be written 



# = 0, ky + fiz = 



•, we have the condition 



Since it also meets line 



I *• 



I °' 



8, 



I o, 



giving A = — 2/a, 



1: # = 



0, 0, =0 



A, ps 



6, —3, 



64, 0, —51 



whence the equations of line 1 are 



2y — z = 



