940 
PROFESSOR A. CAYLEY OR THE SINGLE 
to k-, Jc 7 , k 1(j , Jc n , Zq 3 , k u respectively), while the even functions become equal to c 0 , c lf 
c 2 > ^ 3 > c 6’ c s> c i 2 ’ ^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 Jc’s in terms of (a, h, c, cl, e,f) these 
are proportional to fourth roots, 7 a, &c., 7 ab, &c.; in 7a, a is in the first instance 
regarded as standing for the pentad bcdef and then this is used to denote a product 
of differences bc.bd.be bf.cd.ce.cf.de.clf ef; similarly ah is in the first instance regarded 
as standing for the double triad abf.ccle, and then each of these triads is used to 
denote a product of differences, ab.af.bf and ccl.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 = bcclef, 
ab—abf.ede, 
b=acdef, 
II 
c=abdef 
s. 
II 
§ 
d—abcef. 
ae=aef.bcd, 
e—dbcdf 
bc=bcfade, 
f—cd)cde, 
bd—bdface, 
be=befacd, 
rO 
II 
ce—cef.abd, 
cle=defabe. 
There is no fear of ambiguity in writing (and we accordingly write) the squares of 
7a and Vcib as 7 a and 7 cdj respectively ; the fourth powers are written (7 ci)~ and 
(7ah) 3 ; the double stroke of the radical symbol \ 7 is in every case perfectly dis¬ 
tinctive. 
This being so we have as above c Q =X\/bd, &c., h- 0 = \7 a, &c.: it is, however, im¬ 
portant to notice that the fourth roots in question do not denote positive values, but 
they are fourth roots each taken with its proper sign (+, —, -j -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 x, y are connected with the u, v by the differential relations 
c rdu-\-rdv= — f 
dx dy ] 
7^""77r 
mdv-f pdv= — ^ 
xdx 
71 
? 
