66 ANALYTICAL RESEARCHES CONNECTED WITH [114 
where the symbol <6 (+) is used in order to mark the essentially different character 
of the results corresponding to the different values of the ambiguous sign, then 
be® (—) = /(AX' + bfi + fv') (g\" + ffi" + cv"), 
+ (&/ - ©X' ) {g\" + ffi" + cv"), 
+ (M/V' - f§X") (AX' + V +fv' ), 
+ vW (- M/)+ vKfft + xyy®+X'X" (Z -/jp) 
- ®K-fK. 
= /(AX' + V + >') (g\" + ffi" + cv") 
+ VM(VI© - ffi)(g\" + //' + Cv") 
+ VM(VMS -p|) (h\ f + bfi + fv') 
+/(-M V^+»V^+ffiVllB-MiF-(®»-MJF)) 
= /(AX' + 6/ + //) (gX" + ffi" + cv") 
+ VI (VI© - ffi) (pX" + ffi" + c*") 
+ VM(Vffl - |^) (AX' + V + >') 
-/(VI© - ffi) (VMS - p^), 
that is, &c<b (-) = (VM©-©)(VMS-^){/r£ + VM(r+Z)-/}. 
What, however, is really required 1 , is the value of d>(+); to find this, we have 
be® (+) = be® (-) 4- 2bcK 
=<va«r- ffi) (Vi's - 50 f/zz+va (r+z) +/) 
+ 26ci - 2/(Vlcf - ffi) (Vis - ?B), 
the second line of which is 
2 (VM© - ffi) (VMS - %) (VM© + ffi) (VMS + p^) -/ j 
= -ffi) + ffi)(VMS + ffi)-©ffi + mjfi 
2 (VM© - ffi)(VMS - ffi) VM/ 
1 It may be shown without difficulty that the (-) sign would imply that the sections touching 2 = 0, x=0, 
and x = 0, y = 0, were sections touching # = 0 at the same point. By taking the (-) sign in each equation we 
should have the solution of the problem “to determine three sections of a surface of the second order, the two 
sections of each pair touching one of three given sections at the same point,” which is not without interest; 
the solution may be completed without any difficulty.