704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
499 
69. These are the notations:— 
1. By current-numbers. It maybe remarked that the series was taken 0, 1,..., 15, 
instead oi 1, 2,..., 16, in order that 0 might correspond to the characteristic ^; 
2. By characteristics; 
3. By single-and-double letters; 
4. Gopel’s, in his paper above referred to, and 
5. The same as used in my paper above referred to; 
6. Rosenhain’s, in his paper above referred to; 
7. Weierstrass’, as quoted by Königsberger in his paper “Ueber die Transfor- 
mation der M&eischen Functionen erster Ordnung,” Crelle-Borchcirdt, t. lxiv. (1865), 
p. 17, and by Borchardt in his paper above referred to; 
8. Not a theta-notation, but the series of current-numbers used in Kummer’s 
Memoir “Ueber die algebraischen Strahlen-systeme,” Berl. Abh. 1866, for the nodes 
of his 16-nodal quartic surface, and connected with the double theta-functions in my 
paper above referred to. 
But in the present memoir only the first three columns of the table need be 
attended to. 
70. It will be convenient to introduce here some other remarks as to notation, &c. 
The letter c is used in connexion with the zero values u = 0, v = 0 of the 
arguments, viz.:— 
^0> ^2> %> ^8) $9, ^12) ^15 
are even functions, and the corresponding zero-functions are denoted by 
Cox Ci, c 2> c s , C\, c 6 , Cg, c g , C12, c 15 : 
there are thus 10 constants c. 
When (u, v) are indefinitely small each of these functions is of course equal to 
its zero-value plus a quadric term in (u, v), and we may write in general 
S- = c+i(c ,// , c iv , c y \u, V) 2 : 
there are thus 30 constants c'", c iv , c v . 
The remaining functions 
^7) ^10 > ^11) ^13) 
are odd functions vanishing for 
small, we may write in general 
there are thus 12 constants c', c". 
u = 0, v = 0; when these arguments are indefinitely 
^ = (o', c"$w, v): 
63—2