The first complete proof of the criticality theorem for Bienayme-Galto
n-Watson branching processes was published in Danish in 1930 by J.F. S
teffensen, Here his paper is translated into English, together with an
other proof of the same result, obtained by C.M. Christensen at the sa
me time, These papers are prefaced by a historical review by K. Albert
sen on the problem of extinction of families, also originally publishe
d in Danish.