572 Proceedings of the Royal Society of Edinburgh. [Sess. 
All the expressions for f(a-\-£ 0 ), such as e a+ &, form finite series because 
£o 9 = 0. When we deal with many valued functional forms of f(x), such 
as log x, x h , we have to face formidable difficulties when the general values 
of f((p) are required ; but the difficulties vanish in most useful cases when 
we restrict ourselves in some defined way to one of the many values. 
Thus we will here consider what may be called the principal logarithm 
log 0 0 and the principal square root J 0 <p of (p . 
Guided by (22) and by familiar theorems of ordinary algebra we put 
log 0 <f> = €\og 0 (a + $ 0 ) + r]\og 0 (b + V0 )+ .... ( 
where log 0 (a + 4) = log 0 a + [a" 1 ^ - 0 ) 2 + -....] i ' 
[The principal logarithm log 0 a of the scalar a may be defined in various 
simple ways. Thus putting a = e x+y ' / (~ v > where x and y are real and y is 
between ± 7 r, we may put log 0 a = x + y s /( — 1). Similarly, below, the 
principal square root J 0 a may be understood to mean efta+yv^-i)].] 
By (26) log 0 0 becomes unintelligible when, and only when, one of the 
quantities a is zero ; that is, <p~ x is infinite ; that is, the discriminant of <p 
vanishes. A similar remark applies to the forms below given for <p~ x and 
J o0- 
From (26) it is a simple matter to show that 
= e log0 *, log 0 ./(<£)F(<£) = log 0 /(<£) + log 0 F(</>) 1 
log 0 .</> -1 = -logo <£ - 
Similar to (26) we have 
<f>~ 1 = ((a + 4)" 1 + rj(b + rjo)- 1 + ) 
where ( a + 4)" 1 = a _1 [ 1 - (or^o) + (« _1 4) 2 -••••] J 
L 
Jo<t> = € Jo{ a + €o) + y Jo(b + Vo)+ • • • • } 
where J 0 (a + 4) = J 0 a-[l + \a~ 1 ^ - J(a _1 4) 2 +••••] i" 
. (27). 
. (28) 
. (29). 
The last square bracket contents mean the familiar algebraic develop- 
ment of (1 poT 1 ^)'- 
N.B. — Although when <£ -1 = oo the value here given of J 0 <p is unin- 
telligible, it does not follow that cp has no square root, as is obvious from 
a consideration of the square root of (p 2 ; but I have verified that cases 
occur where there is no square root. 
Clearly (26), (28), (29) are but special examples of a whole class of 
cases. It is perfectly easy, for instance, to write down the form of <p x/y 
where x and y are integers prime to one another, corresponding to the 
form of J 0 cp in (29). 
There is never any question of convergency or divergency to raise 
respecting the series in £ 0 > because they all terminate. 
