on the lately proposed Logarithms of Unity. 283 
From these premises it follows irresistibly that 
ghee {1+ eres + Be Fr wb 
2 Cag 
cist jal alae Opn (6.) 
but by (3.) this product is equal to 
(Y—12wr+1) | (W—12wr+l)? 
esim@L, yal Sips tga 1.2 
x 
i 
(7 —12w2+1) 
1.2.3 
The development (7.) therefore is complete and correct, 
having as many values, and only as many, as e”. 
é Zea, (7.) 
3 End a 21 ’ 
To apply this result, on substituting ee ae for x in 
Qix 1 
(7.), we find that the values of e 2?7—~—1 are represented 
by the formula 
ap (Gv) Lvichaia 
2imr—V/—1 
2war—V/—1\? (7 —1 2i7) 
es eer 1.2 
2wr—V/—1\3 (VW —1 iz) 
ty ™— Vai) 1.2.3 (8) 
Now, among the values of (8.), which are found by giving 
to win that formula all its values in succession, there is one 
Qin 
value (which I denote by e; 2**—“—1) that will correspond 
to the case of w = i, in which case the development (8.) re- 
duces itself to 
(f—12in)  (W=—19ir? W—12in)3 
ie die en ee gag Pc} ON) LE 
= cos(2im)+ /f—I1sin(2inz) = 1. Q.E.D. 
eee 
It may be remarked that, tested by formula (7.), the ex- 
pression ¢ /—12i% wil] be found to have an infinite number 
2G2 
