FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
235 
These successive steps are to be continued until all the cjuantities ^ are determined. 
They, and therefore the numbers x will be determined uniquely, provided that the 
conditions that 
(bi • • ■ V) ] 
(^a+l j 
(^6 + 1, 6+1 j • • • 
&c., 
> shall all be prime to _p, are satisfied. 
When these conditions are satisfied the generators V generate the complete set of 
j 9 -power-exponent numbers. 
(30.) We have not yet seen whether the conditions just found are independent or 
not. We shall find that the first condition includes all the others ; it 
may, however, be replaced by others of a similar kind, but practically simpler. 
Let us write down the complete determinant and divide it into squares and rect¬ 
angles thus:— 
bi 
^12 ■ 
• • '^lo.-l 
ha 
'^lo+l 
hb 
^21 
^22 ■ 
• • h.a-1 
ha 
&C. 
k.-l. 1 
h-l. 2 
■ • k-1.0-1 
^ 0 — 1 . a 
'^a2 
• • L. 0-1 
ha 
^00+1 
hb 
'^a + 1. 1 
• • 
^ 0 + 1 . a 
ki+1. o+1 
'^0+1. b 
hi 
• • 
'bo 
bo+1 
hb 
&c. 
For reference, we may name these squares and rectangles, thus:— 
(aa) 
{ah) 
(ac) 
{ha) 
{hh) 
(he) 
{ca) 
{ch) 
(cc) 
&c. 
2 H 2 
