3^8 Mr. J. Herapath o?i the Binomial Theorem, 



x° 



The natural value is — = oo ; but if this be corrected it be- 

 comes x°-a° [l + (t— 0]°-a° 

 ~ 



0-1 , 0-1 0-2 



l_a°+0(a— 1)+0-— (.r-l)MO-— .— ^ (r-l)3... 







= {x-l) — + -^... 



which is denominated the log. of j". 



We may here observe that the properties of logarithms, and 

 the same form to the function, would be found, if we seek the 

 form of ■\i so that 



•^.x"' may be = n-\)X (H) 



generally. For changing n into + m and adding we have 



y.OT -\-'^.x— = (« + m}\|/jr = \|/,,r — 



or rj/.x" + i}/..r''* = {ii±m)^x^ ^I/.x"±'" (I) 



Putting therefore 3/ = jc" and z = j;"' which, since m and n may 

 be any independent quantities, may allow y and 2 the utmost 

 independence, we find 



tj/j/ + v|/~ = \I/..?/z and \|/?/— \I/~ = vj/. ^ (K) 



Moreover, differentiating (H) we get 



X'if'x=.x^-i/ .x^ ■=■ x°.'i^'.x°=- -i/ A =■ a 



and consequently •^' x = — (L) 



which is the well-known form of the differential coefficient of 

 a X log cr. From the assumed (H) it follows when ?« = 

 that 4/. 1 = ; and the other principal properties of logarithms 

 are evidently contained in (I), (K), (L). 



If we investigate the problem under the still more general 



form of \J/. x^ = n \J/^ x 



and search the relation of the functions \I/, vj/^ we should arrive 



at the same conclusion. 



But to return : if n be a positive integer and /• any non-irv- 



teger, A". O" is finite and a"~'^.0"~' infinite, and of course 

 7ir = 0. Therefore, any non-integral differential of a positive 

 integral power is nothing. This curious property of fractional 

 differentials may be otherwise proved, but my limits will not 

 allow of minutiae. 



If n. be a non-integer and n — r a positive integer, the nu- 

 merator of (9) is infinite, the denominator finite ; and there- 

 fore n ■= CO. It 



