P. HISTORY OF ' AATHEMATICAL - School of Mathematics

274 A HISTORY OF MATHEMATICAL NOTATIONS
more intense abbreviations' sn u for sin am u, cn u for cos am u,
tn u for tang am u, dn u for V (1-k 2 sm 2 am u). The argument K - u,
which is the complement of u, appears in the following abbreviations
used by Guderman (p. 20): amc u=am (K-u), snc u=sin arne U=
sn (K-u), and similarly in cnc u, tnc u, dnc u. Before Gudermann,
Abel 2 had marked sn u by the sign A(IJ). Gudermann's symbols
sn u, en u, dn u were adopted by Weierstrass in a manuscript of 1840,
where he considered also functions which he then marked A(u),
B(u), C(u), D(u), but which he and others designated later' by
Al(u)I, Al(U)2' Al(u)a, Al(u) and called "Abelian functions." In
1854 4 Weierstrass called certain 2n+1 expressions al(uI, U2, ••••)0,
al(ut, U2, ••••)1, etc., Abelian functions; "it is they which correspond
perfectly to the elliptic functions sin am u, cos am u, A am u." He
shows the relation" al(u1, U2, •.••). = Al(uI, U'l, ••••). : Al(Ut, U2,
....). In 1856 he changed the notation slightly" from al(uI, U2 ,••••)0
to al(uI, ....)1, etc.