.934 
PROFESSOR A. CAYLEY OR THE SINGLE 
, = B-(0, q) C 2 (0 ,q) A(0, gQD'(0, q) 
l ' A 2 (0 ,q) 1 A°-(0 ,qj V B(0, ? )C(0,y)’ 
7 C 2 (0, r) , B 2 (0, r) A(0, r)D'(0, r) 
1 A 2 (0, r)’ A 2 (0, r)’ B(0, r)O(0, r) ’ 
1 ^ 1 
log (7= ——— > low r= — 
t=> 1 j v O 
7tK 
- ? 
K' 
where if the identity of the two values of k or of the two values of k' were proved 
independently (as might doubtless be done), the required theorem (r the same function 
of k' that q is of k) would follow conversely: and thence the other equations of the 
system. 
G6. We have in the ‘ Fundamenta Nova,’ p. 75, the equation 
H( m, k) 
0 ( 0 , k) 
H (u, k') _ 
0 ( 0 , 7C) ; 
writing here K 'u instead of u the equation becomes 
H(&, k) 
0(0, k) 
H(K ’u, k‘) 
0(0, k') 
or what is the same thing 
(- 
log q) 
~D(u, r) 
' C(0, r) 
which can be readily identified with the foregoing equation between D 
au 
771 
and 
T)(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, &c. 
67. We have here 16 functions 3- 
, v) : this notation by characteristics, con¬ 
taining 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 correspondence 
of the two notations, nor is there anything in the definition of the functions which in 
