(427) 
when the s symbols ge) and p‚@k-D) represent polynomia of 
degree n—k—i in 4. From this ensues by integration 
ai = (t—t) A+! pin—k-1)_b,, (Cay SP ae MV (t—t,)**) york) + Dy, 
in which the quantities 0; and 5, denote constants. So tne trans- 
formation of coordinates to parallel axes characterized by the formulae 
b; 
eee heee a) 
bo 
finally gives 
(t—t,)* +1 zi -k—1) 
fi? Vv 
gi 
2 ree 8) 
by which we alight on the case that the s equations «;= 0 be- 
longing to (1) have a common & + 1-fold root 4, whilst » moreover 
after being diminished by a constant quantity b, is divisible by 
(t — tk 
The particularity treated here appears only in the case when 
n . . . 
the curve As has singular points of a definite character. So 
the simplest case of a common double root t, of the s equations 
i=0 implies that the origin of each of the spaces of coordinates 
E;=0 represents two of the » points of intersection with Rs , which 
with a view to the equality of the values of the parameter belonging 
. -y ft . 
to those points only then takes place when #; shows a cusp in 
this point. We see at the same time that we have not generally 
enough enunciated the case sub 3%). For from this appears that the 
particularity will come in as soon as As has a cusp anywhere. So 
the case sub 36) ought to run: “The equations «; = 0, (i= 1,2,...3) 
have common equal roots or a transformation of coordinates to 
parallel axes can call forth this particularity.” 
Of course the case may present itself that ¢ 1s a common equal 
root of the s equations /7;= 0, but that the degree of multiplicity 
in relation to those equations differs. If t is a 4j-fold root of (7; = 0, 
a ky fold root of 73 = 0, ete, then for k we must take the smallest 
number 4;. 
If it happens p times that a transformation of coordinates to 
parallel axes implies the particularity indicated here, and if 
ky, kg,...k, are the smallest numbers £ for. each of the correspond- 
31* 
