sc sb whee Ori ee oe 
47 
we here again have a case of 0 x o, correctly interpretable 
by the left hand member of the equation ; that is, the right 
hand member, when z = 0, is accurately 1. When «2 = =, the 
- signs of the series all become minus: therefore the true value 
- jn that case is — 1. 
Before concluding this subject it may be proper to observe, 
that the investigation, whence the series for 5 is usually de- 
duced, is deficient in generality. Whenever logarithms are 
employed in connexion with imaginary quantities, the imagi- 
nary forms of the logarithms, as well as the real, ought always 
to be introduced into the investigation : hence the logarithmic 
expression, from which the series alluded to is derived, should 
be written thus: 
ge 2 Tas, ee 
log u=u—w-— = ee va + Be. + hry I 
By substituting in this e* V—1 for u, and then dividing the re- 
sult by 2Y —1, we shall have the correct and general form, 
Bis es sin2z2  sin32_ sin4z 
9 ig 2 3 4 
where k is any whole number, positive or negative, deter- 
minable in any particular case, so as to conform to the first 
member of the equation: regarding that first member, # not 
exceeding 7, as indifferently either > or kr + > 
+ &e. + kr, 
I have here used the limited logarithmic forms of Euler, 
and not the more general ones furnished by Mr. Graves’s 
theory of imaginary logarithms,* since these limited forms 
are sufficient for all the real values in the general result. 
_ It now merely remains to be shewn that, as briefly stated 
at page 43, the differential theorem is inapplicable, not only 
when the proposed series is divergent, but also when it ceases 
to be convergent, and becomes what Hutton has called a neu- 
* Philosophical Transactions, Part I. 1829. 
