115 
for upon solving the equation (s — a) (s - b) = c' 2 , it is easily 
seen that one of its roots a is greater than both a and b, and 
the other /3 is less. Therefore, since + co , a, /3, — co when 
substituted for 5 give results + , —, +, —, there is one root 
greater than «, another between a and /3, and a third less 
than /3. 
141. Every variety of surfaces of the second order referred 
to rectangular co-ordinates is comprehended, without exception, 
in the equation 
Ax 2 + By 2 + Cz 2 + 2 A’oc + 2 B’y + 2 Cz + D = 0. 
Let the surface be represented by the general equation of 
the second degree f (x, y, z) = 0, and let it be referred to 
three new rectangular axes Ox', Oy, Oz', by the substitutions 
of Art. 90, and let the transformed equation be 
Ax' 2 + By' 2 + Cz’ 2 + 2 A'y'z' + 2B'z'x' + 2 Cx’y + 2 A"x' 
+ 2 B"y’ + 2 C"z r + D = 0; 
also let the equations to a line parallel to a system of principal 
chords (the existence of which in every case is certain) referred 
to the new axes, be x = mz , y'—nz, then m and n satisfy 
the conditions 
Am + C'n + B' = m (C + B’m + A n) 
Bn + Cm + A' = n (C + B’m + A'n). 
Suppose now one of the new axes, that of z for instance, to 
be parallel to the direction of the principal chords ; therefore 
m = 0, n = 0; consequently we must have A' = 0, B'= 0. 
Hence whenever one of the rectangular axes is parallel to the 
direction of a system of principal chords, the general equation 
is freed from two of the rectangles, and takes the form 
Ax 2 + By' 2 + Cz 2 +2C'xy'+2A”x'+2B''y -\-2C' z'+D=0,..(2). 
This equation comprehends all surfaces of the second order 
without exception, and it may be still further reduced if, 
without altering the axis of z', we turn the axes of x and y 
in their own plane through an angle 0, so as to make the 
term involving x'y' disappear, by the substitutions (Art. 94.) 
8—2