( 501 ) 
property *), according to which the singular elements (branch-elements 
and double-elements conjugate to them) of a correspondence (2,2) 
can be arranged in such a way that the singular elements of the 
first system correspond projectively to those of the second one. 
For, if two pencils are connected by a (2,2) we have but to 
rotate them around their vertices until a branch-ray of the first 
pencil coincides with a branch-ray of the second; in the new 
position they then generate a C, with two cusps. From this is 
evident that there are four projectivities between the singular elements *). 
The (2,2) between the pencils «,—= Ar, and «,—=wuea, has as 
equation 
| Au + 2u + 26,44 2bu+c=— 0. 
By the points of 4 these pencils are arranged in the projectivity 
6,4 = bi! 
By eliminating 2 we find out of these two relations the equation 
of the correspondence (2,2) between the points which conjugate 
rays of the pencils (O,) and (O,) generate on h. And now it is 
evident from 
bu? aw? + 2b, by uw + 2b,°b, (ut uw) + 6% c=0 
that this correspondence is involutory. 
This result is in accordance with the well-known property *), 
according to which a (2, 2) between two collocal systems is involutory 
when the two systems have the same branch-elements. 
3. Evidently the involutory (2,2) on A does not differ from the 
(2,2) which was deduced from the fundamental involution F,. Its 
coincidences arise from the four tangents which one can draw from 
H to C,. Indeed, the polarcurve of H consists of the line h and 
the conic u (passing through the points of contact of d). 
If the branch-point R=vr,r, is conjugate to the double-point 2’, 
then A’ must be the point of intersection of the rays which the 
points of contact R, and R, of r, and-r, project out of O, and O,. 
We conclude from this that the tangential points R,, S, T, of the 
1) Emm Weyr, Beiträge zur Curvenlehre, Vienna 1880, Alfred Hölder, p. 32, or 
Annali di Matematica, 1871, IV, p. 272. 
3) In my paper “Over vlakke krommen van de vierde orde met twee dubbel- 
punten’ (N. Archief voor Wiskunde, 1888, XIV, p. 193) I have applied the pro- 
perties of the (2,2) correspondence to those curves. 
3) Emm Weyr „Ueber einen Correspondenzsatz’, Sitz. ber. der K. Akad. in 
Wien, 1883, LXXXVII, p. 595, or my paper under the same title in N. Archief 
voor Wiskunde, 1907, VII, p. 469. 
