( 533 ) 
The equation of the pencil of quadratic surfaces now becomes 
ff led itn eerie ease AR) 
where however A and 4 have a wider meaning than before, A being 
ant? + a,,y* + a,, 2? + 2a,, ey + 2a,, wz + Za, ye + 
+ 2a,,u + 2a,,y + 2a,,2+4,,, 
and 5 being the same expression with the coefficients 5. 
Let us now put the coordinates of the centre of the surface (1) 
pqr and let us regard this centre as origin ©’ of a new system 
of coordinates with axes parallel to the original ones. We then 
arrive for surface (1) at an equation in #’, 4’, 2’, in which the terms 
of order one are missing and those of order two possess the same 
coefficients. The principal axes of this surface are given by the three 
equations : 
(a, a! J G4 y + a, 2) + À (b, a’ + Bis y' + 6,, 2’) + ka’ = 0, 
(a,,0' + a,,y' Hans) + 4(b,, el +5,,y' + b,,2') + ky’ =0, 
(a. T+ a3 y =e z) +4 (bs aw ban y + b,, z') hei: 
As could be foreseen the elimination of 2 and # furnishes the 
same equation as was already found for the cone of axes. 
If we wish to form the equation with respect to the original 
system of axes, we must put «’ =2—p, y’=y—q, 2 =2- Pr 
and make use of the equations of condition for p, g, 7: 
(a, ie Àb)p == (a4, a= Ab,,) q + (a; ia àl) r + dis oi DD = | 
(41, Sin Ab,,)P + (ass + db) q + (4,3 + Ab) Te dan ie db, = | 
! 
(as aH Ab) p hi (a; in Ab) q si (an ie Ab55) r “= Cs 4 + abs, = 0. 
By this substitution the equations assume the following form: 
(aje Haay O52 + 4) HA (bit + Drag + bne Hb) + k(e-p) = 9, 
(eo a Ayo = daz =F d,,) si À(be ie Day se boa? = ba) “5 k(y-q) = 0, (3) 
(4,50 + day + 4552 + 4,4) + 4(0,,% + 0,,y + 65,2 + Das) + Kler) = 0, 
or written shorter 
A, + B,~+k(@—p)=0, 
Ap ot see Ane t= (ie Q) k=" Ose eee weet ator eae (4) 
A, + B, +k (2 —r)=0. 
The surface S, is obtained by eliminating p, q, 7, k, à out of the 
equations (2) and (4). 
4. This elimination leads to extensive ealeulations as the variables 
appear also as products two by two, We shall here point out the 
