176 
MR. GRAVES ON A RECTIFICATION OF THE 
consisting generally of discontinuous points, can accurately be called curves), 
whose equations involve exponential functions. By their means also, various 
differential and other formulae usually exhibited in logarithmic treatises maybe 
rendered complete. An extended pursuit of these objects would exceed the 
limits of the present design ; but to explain briefly the mode of procedure 
employed in application of the preceding general results, an Appendix is sub- 
joined, containing a few examples. 
APPENDIX. 
^ 1 . The constant c might appear to be needlessly introduced, if its necessity to 
insure the convergence (and universal accuracy [Note E.] ) of the 
series [18] were not plain from what follows. 
Differentiating n terms of the series [18] there results, 
+ (1 -Qfc)... + (1 - flf c) n ~ l ~ydb 
which, as is evident on multiplying by 1 — (1 — die), 
- - V~=lfc } d9 or [24] 
This expression, if the series [18] be convergent, or, carried to infinity, 
be numerically equivalent to f -1 0, ought, as n is increased without limit, to 
approach indefinitely to d f -1 6, or (see [13]) — 1 0) _1 d 6 ; but, on referring 
to [24], it is obvious that such can be the case only where c is so assumed, 
that, n being supposed to increase without limit, (1 — Hc) n shall approach in- 
definitely to 0. 
Were c neglected, or, in other words, taken = 0, and therefore (see [6] ) 
f c— 1 , 0 would not always necessarily lie between such limits that ( 1 — 6) ” 
should possess this property ; but a quantity f c is, in any case, supposable, 
which will insure for ( 1 — 6 f c) n the required essential, whatever, at the time, 
be the value of 6. 
§ 2. If a have among its values two quantities differing only in sign, x must 
be a rational fraction with, in its lowest terms, an even denomi- 
nator. [Note F.] 
By [20] all the values of a are expressed by f («r f -1 a). Any determined 
