34 DR. A. JOHNSON ON SYMMETRICAL 
Two values of # may therefore be found enabling us to satisfy (8). Now it is easy to 
show that these values are the co-efficients of Y’ and Z in (5). For multiplying the equa- 
tions of (7) by à, b, ¢,, respectively, and adding, we get by condition of perpendicularity, 
i (ab ot) = 4} a, CB 5,22 
d a, 
3 
das 

ae 
2 ie 
= fo (Az bg 6s). 
since the function is homogeneous. 

Hence 
_ hf (as bs ¢3). 
= re 
and we must also have, obviously, 
Fo (Gy by C2) 
Ce —- RE 
For 
| d fe d fo d fr 
da Enr Gr RE 2 
is the same as 
Gh ify Oe hips Gifs 
7 3 
dCs 

and hence equations (6) will enable us to find # as a function of either 
Qs, On, Cz, OF Of Az Dz, Cy. 
The equation of the section is thus reduced to 
hythe=l (11) 
(using small letters for the co-ordinates now) where 4, and #, are the roots of (10). We 
val à 5 ’ 
see thus that the two values of x are the squares of the semi-axes of the section. 
v 
To find the equations of the axes of the section. 
We can, of course, put these in the shape 
DU z 
a PAT: 
x VEN DRE 
Ah ly T 
where a, and a,, etc., are the values that satisfy the first three of equations (8), when the 
two values of & are successively inserted. 
These will, however, be subject to the condition given in the fourth equation. 
The actual equations of the axes, free from any condition, can be easily found after 
stating the two following preliminary propositions : 
