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 formulee usually exhibited in logarithmic treatises may be 

 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 accui-acy [Note E.]) of the 

 series [18] wexe not plain from what follows. 



Differentiating n terms of the series [18] there results, 



--/":^fc/l +(1 -«fc)... + (l -fifc)""'\rf9 

 which, as is evident on multiplying by 1 — (1 — ^fc), 



= _^Tnf,{L^il^^}d9 or (^-irifl)-^|i_(i_flf,)»J,d9[24] 



This expression, if the series [18] be convergent, or, carried to infinity, 

 be numerically equivalent to f ~ 6, ought, as n is increased without limit, to 

 approach indefinitely to d f" 6, or (see [13]) {^ — 1 ff)~ d ^ ; 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 — ^fc)" shall approach in- 

 definitely to 0. 



Were c neglected, or, in other words, taken = 0, and therefore (see [6]) 

 f c = 1, 6 would not always necessarily lie between such limits that (1 — ^)" 

 should possess this property; but a quantity fc is, in any case, supposable, 

 which will insure for (1 — ^f c)" 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 i{xi~^ a). Any determined 



