Mb. DE morgan, ON DIVERGENT SERIES. 191 



SECTION III. 



It generally happens that the real analytical equivalent of the different values of an 

 indeterminate expression, is the mean of those different values. 



This principle must rest at present upon induction. When Leibnitz pronounced 1 to be the 

 value of 1 — 1 + ... because there was no apparent reason why either 1 or should be preferred, he 

 was not only right in his conclusion, but had a glimpse (though not in solid reasons) of a principle 

 which admits of such frequent confirmation that it may be suspected to be general. 



In the first place, if we take any algebraical series, such as a + bsc - c.v^ + ax^ + bx* - cr^ + ... 

 in which c = a + /;, so that when x = 1, the successive results of summation are a, a + b, 0, a, a + b, 

 0, &c. we find by common processes that the analytical equivalent is the mean of o, a + b, 0, or 

 l(2a + 6). The same thing happens if we take other forms which produce the same limiting form, as 

 a + b cos 6 - c cos 29 + ... 



Secondly, if we take a series A„ + A^cosG + /^cos 29 + ... or Fourier's integral J^ f_^ cos 

 w {x — v) (hv dtv dv, in such manner that it may represent the ordinate of a discontinuous curve, 

 the branches of which do not join at the common ordinate, it is found that for the abscissa of the 

 common ordinate the series and the integral represent in both cases, not either or both of the 

 ordinates, but the mean between them. 



Thirdly, the indeterminate symbols sin oo and cos eo are found in numberless cases to represent, 

 each of them, 0, the mean value of both sin.r and cos.r. The mean value of any function chx, 

 between a and b, is J^ (pxdx divided by b - a. 



Fourthly, if m lie between — / and + I, Poisson has shewn that 



1 /- + ', , 1 V- f r + ' mirCx-v) ^ , 1 -, 

 <px = — j (pvdv + -2, < I cos . (pvdvy (from m = 1 to r» = 03 ), 



the second side of which is not changed in value, by changing the sign of /. And this second side 

 is the same whether we make x = — I, or x — + I ; consequently it is wholly undecided whether it 

 is then to represent (p{- I) or (p{l). Poisson has shown that in either case it represents the mean 

 of <^(0 and (p{-l). 



Fifthly, if we extend the term mean value, and, in cases in which the function becomes infinite, 

 define it as fl(pxdx-~(b - a), the same principle applies, in a very peculiar manner, to the remaining 

 trigonometrical functions^ if the part of the integral at which (hx becomes infinite, be examined in 

 the manner which occurs so frequently in the writings of M. Cauchy. Let us take for instance, 

 tan tr. In fg" tanxdx, the finite parts destroy one another: and to obtain the expression for it we 

 must examine the integral from Att-ju, to ^ir + n, and from ^tt - m to |7r + /i, /i being infinitely 

 small. Now the indefinite integral is - log cos x, so that we have to examine 



log e^iii^L^ and log ^_^i(tZLZ^ 

 ° cos (^TT + yu) cos (f 7r + m) 



each of which is log (- 1) or ttv- 1, when /i = 0. Hence f^ tan xdx is Sttv- I, which divided 



by Srr, gives v - 'i 'he proper representation of tan eo , if this principle be true. Now if we 



1 • , ^ tan w + tan « , , . „ . . , 



examine the equation tan (.r + «) = ^ and make x infinite, presuming that 03 and 



^ ^ ^^ 1 - tan iS tan y ' " 



« + y are the same angles, we find tan eo = i'x/- 1. In the same manner cot oo is ± v^- 1. 



It cannot be argued that since the values of tan x, from * = to x = w, have signs contrary to those 



from .r = TT to .x; = 27r, therefore if it \/ - 1 be taken for tlio first, - ttv- 1 should be taken for 



the second : the reason being that the signs in the second semicircle are really repetitions of those in 



