190 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
[665 
the differential equations which connect u, v with x, y are 
dx dy 
du ~ ^/X + VF’ 
? xdx ydy 
0 VF* 
There are here sixteen ^-functions A, B, C, D, E, F, AB, AG, AD, AE, BG, BD, BE, 
GD, GE, DE, and an associated ©-function ft, where for shortness I use the single and 
double letters A, B, AB, ...,ft, instead of functional expressions A (u, v), B (u, v),.., 
AB (u, v),..,£l(u, v), to denote functions of the two letters u, v. Writing also 
(a, b, c, d, e, f) for the differences a — x, b — x, etc., and (a 1( b 1} c 1} dj, ej, f,) for the 
differences a — y, b — y, etc., whence VX = Vabcdef and 'dY — VajhiCidjejlb and 6 for 
the difference x — y, we have sixteen ¿¡^-functions which are represented by 
da, \Jb, \Jc, \Jd, sje, df dab, Vac, Vad, Vae, dbc, dbd, dbe, dcd, 'dee, Vde, 
the values of which are 
* 
y'a = Vaa 1 , (six equations), 
dab = ^\dabfc 1 d 1 e 1 — VaJbjbcde}, (ten equations), 
and the definitions of the sixteen ^-functions and the ©-function are 
A — ft Va, (six equations), 
AB = ft 'dab, (ten equations), 
and one other equation to be afterwards mentioned. 
I call to mind that, in a binary symbol such as dab, it is always f that accom 
panies the two expressed letters a, b: the duad ab is, in fact, an abbreviated expression 
for the double triad abf.cde: and I remark also that I have for greater simplicity 
omitted certain constant factors which, in my second paper above referred to, were 
introduced as multipliers of the foregoing functions *Ja, ...,'dab,... I remark also that, 
to avoid confusion, the square of any one of these functions Va or dab is always 
written (not a or ab, but) ('da) 2 or (dab) 2 . 
I use 0 as a symbol of total differentiation: thus 
0A = du + ^ dv, d 2 A = (0m) 2 + 2 ^ (0m dv) + (^)“> etc. 
du dv du 2 dudv dv 2 
Moreover I consider du and dv as constants, and use single letters \, L, etc., to 
denote linear functions udu + ¡3 dv, or quadric functions a (du) 2 + 2/3 du dv + y (dv) 2 (as the