SURFACES OF THE SECOND DEGREE. 91 



also the sign of V^, a,^ may be substituted for Vi. In this case, (10) 

 and (9), V2 has the form 



P+ Q + 2? + 2\/QK" cos ?+2\/;BP cos .7+2 ^/PQ cos ^, 



which, when P, Q, and R have the same sign, is always of that sign; 

 and therefore can only be = when P, Q, and B are severally = 0. 



When ^"3=0, and F'i = 0, in which case W appears in the form -, 



and its real value is W (27), the simplest criteria of which are ex- 

 pressed in (26) the equations (30) and (34) assume the forms 



Aaf' + A'y"+Jr'=0 (42), 



KA'- r,A + r,=o (43), 



the first of which, if V^ be positive, and F", and W of the same sign, 

 is impossible, and belongs to an elliptic cylinder if V^ be positive, 

 and Fi and W of different signs. As before, we may substitute a„ 

 for Vi. If V2 or a^i be negative, (42) belongs to an hyperbolic cylinder : 

 and if V2 — O, in which case a^^ = 0, h,i = 0, and c^^ = and W is infinite, 

 we have a parabolic cylinder. It appears therefore, that any surface of 

 the second order, which has three parabolic sections, not having a 

 common line of intersection, is a parabolic cylinder. The central line 

 of this surface is at an infinite distance. When W' = and V is 

 positive, equation (42), considered as of two dimensions, represents 

 only the origin, and therefore belongs to a straight line, the axis of 

 iB'. When Fa is negative, W being =0, (42) is the equation of two 

 planes intersecting at an angle whose tangent is 



2^/-AA' 2V-F,r, 



A + A' ' °^ r, 



When the equations of the center belong to a plane, and W as 

 well as W appears in the form -, the real value of W is W", given 

 in (29) and the simplest conditions are, as in (28), 



«// = *// = C// = 0, 



a : b : c '.: a : c : b. 



M 2 



