66. We have, in the Fundamenta Nova, p. 175, [Jacobi’s Ges. Werke, t. i., p. 227], 
the equation 
© (0, lc) v k' 0 (0, k') 5 
writing here K'u instead of u the equation becomes 
H(iK'u, k) . /k (. ttK' \ H {K'u, k') 
■@(0, k) =' vfc' exp -1“) ■ -0(07W' 
or, what is the same thing, 
C(o, , 
which can be readily identified with the foregoing equation between D and 
B(u, r). But the real meaning of the transformation is best seen by means of the 
double-product formulae. 
THIRD PART.—THE DOUBLE THETA-FUNCTIONS. 
Notations, Sc. 
67. We have here 16 functions ^ gj ( u > v ) ■ notation by characteristics, 
containing each of them four numbers, is too cumbrous for ordinary use, and I 
therefore replace it by the current-number notation, in which the characteristics are 
denoted by the series of numbers 0, 1, 2,..., 15: we cannot in place of this introduce 
the single-and-double-letter notation A, B,..., AB, &c., for there is not here any cor 
respondence of the two notations, nor is there anything in the definition of the 
functions which in anywise suggests the single-and-double-letter notation: this first 
presents itself in connexion with the relations between the functions given by the 
product-theorem: and as the product-theorem is based upon the notation by charact 
eristics, it is proper to present the theorem in this notation, or in the equivalent 
current-number notation: and then to show how by the relations thus obtained 
between the functions we are led to the single-and-double-letter notation. 
68. There are some other notations which may be referred to: and for showing 
the correspondence between them I annex the following table:—