380 Professor Hami.ton on Conjugate Functions, 
divided, by distinguishing between odd and even values of the positive or contra- 
positive whole numbers. or, if we suppose that B denotes a positive ratio, and that 
m and n denote positive whole numbers, we may then suppose 
b=8, or 6=0, or 6=0 8, 
w=M, or n=O, or p=O m, (296.) 
v=n, or w=0; or p= O0in; 
and thus shall obtain the twenty-seven cases following, 
” 2 On 
B™ B*”, Bm 
ze, g pa 
Be 5 Be ’ B o (297.) 
n mos on 
BeOm, BOm, Bom 
On 5 O m ; Om 
n oO on 
Oe Dan Ur ois (298.) 
n o on 
Qem, O om, O om, 
2 ws ous. 
(9 B) my (O B) my (9 B) my \ 
n o on 
(9B)*, (OB)*, (OB) -, (299.) 
ne Qe en 
(9 B) on, (9 B) om, (9 B) em, J 
which we may still farther sub-divide by putting m and m under the forms 
m= 2 21, \or M=0 1 +.2.2, 
n=2k, or 1=O0 142k, (300.) 
_in which @ and & themselves denote positive whole numbers. But, various as these 
- cases are, the only difficulty in discussing them arises from the occurrence, in some, 
of the ratio or number 0; and to remove this difficulty, we may lay down the fol- 
lowing rules, deduced from the foregoing principles. 
To power the ratio 0 by any positive whole number m, gives, by (293.), the ratio 
O as the result. This ratio 0 is, therefore, at least one m’th root of 0; and since 
no positive or contra-positive ratio can thus give 0 when powered by any positive 
whole number, we see that the on/y m’th root of O is O itself. Thus, 
1 
0m=~0, (301.) 
acne tw 
Pa 
4 
° 
