( 536 ) 
bis = 0; -b,,—0; %,,=0; 6,,=0, whilst also“, Br 
simple forms and all coefficients except Q and Q’, are simplified. 
At the same time this substitution shows that in equation (8) a 
factor may be omitted; if namely we make use of the above named 
values for bj, we shall find: 
Vib ld 5 VNS 12 
from this ensues that the first column of the determinant (8) is 
divisible by £,. This divisibility is connected with the fact that the 
equation of the locus of the axes must become of order nine, whilst 
when developed the determinant (8) becomes of order twelve. So 
when a complete operation is executed factors must disappear out of (8). 
4. Out of the former geometrie treatment it is evident, that in 
some cases the locus of the axes S, breaks up. As one of the special 
cases appearing there the case of a cireular base curve of the pencil 
was treated where S, broke up into a cubic surface and into a 
surface of order six. The equations of the algebraic treatment of this 
case become, when one chooses the plane XO Y as the plane that 
is intersected according to a pencil of circles: 
‚2 PPM 1) ET) 5 Or 
A=a,, 27 + a,, y” + 2a,, vz He 2a,, ye 2Q,, 2 Arn 
B= bss 2? + 26,, ez + 26,, ye + 26,,2 + 20,, 2 =0- 
From these equations the simplified values for M, V.... can be 
deduced. 
(February 23, 1905). 
