663] 
177 
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
and these are all included in the equations (10), (4), (12), (15), (6), which serve to 
express G, B, E, F, I in terms of D, H, G, A, J, i.e. ac, ce, eb, bd, da in terms of 
ab, be, cd, de, ea, if for the moment we write G = ac, etc. But the five linear relations 
in question are, it is at once seen, satisfied by A, B, J considered as given functions 
of a, /3, y, 8. 
The equation VAJ ± VBF ± VCG = 0, substituting for A, B,...,J their values in 
terms of a, b, c, d, e, f becomes 
Vabc. def. aef. bed, + Vabf. ede. ace. bdf + Vabe. cdf. acf. bde = 0, 
which (omitting common factors) becomes Vbe-. ef - ±*Jbf ' 2 . ce~ ± Vbe 1 . cf- = 0 ; or, taking 
the proper signs, this is the identity be . ef+ be .fc 4- bf. ce = 0. 
It is to be noticed that 
8- + a 2 - /3 2 - 7 2 , 2 (a/3 - 78), 2 (7a + ¡38), 
2 (a/3 + 78), 8' 2 + /3 2 — 7 2 - a 2 , 2 Q3y — a8), 
2 (7a — /38), 2 (Py + ag), 8- + 7 2 — a 2 — /3 2 , 
each divided by 8 2 + a 2 + /3 2 + y 2 , form a system of coefficients in the transformation 
between two sets of rectangular coordinates. We have therefore 
\/ab, \/ad, Vce, 
'dbe, ^ de, \/ac, 
V be, V cd, V ae, 
each divided by \fbd, and the several terms taken with proper signs, as a system of 
coefficients in the transformation between two sets of rectangular axes: a result which 
seems to be the same as that obtained by Hesse in the Memoir, “ Transformations- 
Formeln fur rechtwinklige Raum-Coordinaten ”; Grelle, t. lxiii. (1864), pp. 247—251. 
The composition of the last mentioned system of functions is better seen by writing 
them under the fuller form ^labf.cde, etc.; viz. omitting the radical signs, the terms are 
abf. ede, adf. bee, abd. cef, 
bef. acd, def. abc, acf. bde, 
bef. ade, cdf. abe, aef. bed, 
each divided by bdf .ace; or, in an easily understood algorithm, the terms are 
bf.d df.b bd.f 
a.ce 
bf. d df. b bd .f 
e.ac 
bf.d df.b bd.f 
c.ae 
bf. d df. b bd .f 
each divided by bdf. ace. 
C. X. 
23