704] A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
503 
form of o), the odd functions each vanish (their evanescent values being proportional 
to k 5 , k 7 , k 10 , k n , k 13 , k u respectively), while the even functions become equal to c 0 , c u 
c a , c 3 , c 4 , c 6 , c 8 , c 9 , c 12) c 15 respectively. 
Observe further that on interchanging x, y, the even functions remain unaltered, 
while the odd functions change their sign; that is, the interchange of x, y corresponds 
to the change u, v into — u, — v. 
77. As to the values of the 10 c’s and the six k’s in terms of a, b, c, d, e, f, 
these are proportional to fourth roots, y/a, &c., v 7 ab, &c.; in VZ, a is in the first 
instance regarded as standing for the pentad bcdef, and then this is used to denote 
a product of differences be .bd .be .bf .cd. ce. cf. de. df. ef\ similarly ab is in the first 
instance regarded as standing for the double triad abf. ede, and then each of these 
triads is used to denote a product of differences, ab.af.bf and cd.ce.de respectively. 
The order of the letters is always the alphabetical one, viz. the single letters and 
duads denote pentads and double triads, thus: 
a = bcdef, 
b — aedef 
c = abdef 
d = abcef 
e = abedf 
f = abede, 
ab = abf. ede, 
ac = acf. bde, 
ad = adf. bee, 
ae — aef. bed, 
be = bef . ade, 
bd = bdf. ace, 
be = bef. acd, 
cd = cdf. abe, 
ce = cef. abd, 
de = def. abc. 
There is no fear of ambiguity in writing (and we accordingly write) the squares of 
y/a and ab as Va and Vab respectively; the fourth powers are written (Vaf 
and (v/ab) 2 ; the double stroke of the radical symbol is in every case perfectly 
distinctive. 
This being so we have as above c 0 = '\.y // bd, &c., & 5 = A,v / c, &c.: it is, however, 
important to notice that the fourth roots in question do not denote positive values, 
but they are fourth roots each taken with its proper sign (+, —, +i, — i, as the 
case may be) so as to satisfy the identical relations which exist between the sixteen 
constants; and it is not easy to determine the signs. 
The variables x, y are connected with u, v by the differential relations 
a du + rdv = — ^ 
j dx dy ) 
WX VF)