36 DR. A. JOHNSON ON SYMMETRICAL 
Taking then the solution of the system of equations as 
© y z w 
AU. GA TT 
we find it is equivalent to 


a ie 2s y Z 
VA BP) NV EN 
19 
Diy BEV CNED. 

or 



Symmetrical equations of the axes. 
Applying equations (14) to the solution of equations (8), and squaring to remove the 
radical, we get 


ie C3" ae i’ 
b—k, 1, CA) PT ae eee CA] > TANT m (15) 
l, e—k, «a, mn, a—k, a, RD oy n, b—k, it 
b, Ci Ci ay ay b, 




Substituting +, y, 2, for as, b, ¢, in the first three of these we have the equations of the 
axes when the two values of & are inserted. 
It is convenient to note here that the value of h may be found from equations (7) in 
the following form : 
Multiplying the equations by 4, b,, ¢, respectively, adding, and solving, we find 



af oh i d fo 1 
=e 2 2 £ 16 
naa (a 4,58 + a Tr af + 07 + c? ( ) 
From symmetry we see also that 
d fs d fr d fs 1 + 
= i LUE Ile 
hat (a Gh Sey oO ae + be + ee (17) 
Geometrical interpretations. 
1’. The geometrical interpretation of the first of the conditions (6), viz: 


df, Gehan, VO 
Ta ae ae 
is that either of the axes of the section, viz: 
‘ ei 
Bc ip Bet a pee ne 
Ay ba © As b, C3 
lies in the diametral plane conjugate to the other (in other words, that the axes are a pair 
4 as 
of conjugate diameters), for the diametral plane conjugate to —= =: 

fr d fe a fo 
Sera be dy EU 


