m 



wliicli is described by the pair X, Y, will be the combination of 

 twice «^ tive times a.^' and twice {h^. 



Hence 



8^; = 16 + 5.r 4- 18i/ : . (7) 



If Z describes the curve Vl^ the corresponding figure of order 4s 

 consists of the curve /?i% of three times a^, and of the 8 curves 

 ^3gi{k::^\). Hence: 



4^ = 4 + 3a' + 83/ (8) 



Out of (6), (7), (8) we find by elimination of x and ?/, 

 z^- — 11 z + 882 = 0; 

 so z is equal to 63 or 14. The second value, however, must be 

 rejected ; for we have proved above, that [XYZ) is of the class 21, 

 so that / has 42 points in common with I at the least. So we find 

 the values 



z = 63, K r= 40, y ■=. \Q). 



For the involution [XYZ), ^ is a singular point of order 40, 

 Bk a singular point of order 16. 



As / and ). besides the 21 pairs already mentioned can onl}"^ have 

 coincidences in common, the curve of coincidences {XYZ) is of 

 order 21, ó'\ 



Apparently «^° has in A a 20-fold point, /^a-J" in Bk an eight-fold 

 point ; in these points <P^ has the tangents in common with «^" and ^^.^^ 



If A' is placed in A and Y in B^, x=zX' X" envelops a curve 

 of the 5''' class, // = Y'Y" a conic; so there are 10 straight lines 

 X ■= y. From this it ensues that the singular curve «^° has ten-fold 

 points in Bk- In a similar way we find that the curve /r^-^" has 

 qnadnijilc i)oints in 7>/; it passes ten times through A, eight times 

 through B].. 



Mathematics. — ''On the functions of Hermite." (Third part). 

 By Piof. W. Kapteyn. 



(Commimicaled in tlio meeting of May 30, 1914). 



12. After having written the preceding pages, we met with two 

 important, newly published papers, on the same subject. The first b}' 

 Mr. H. Galbrln: "Sur un développemeut d'une fonction a variable 

 réelle en série de polynomes" (Bull, de la Soc. math, de France 

 T. XLI p. 24), the second by Prof. K. Runge ' L^eber eine besondere 

 Art von Integralgleichungen" (Math. Ann. Bd. 75 p. 130). 



