= - oN a 
process of incan values giving an ambiguous result, such as + V—1, would 
imply that the real result is the mean between the two values of the ambi- 
guous expression, which in the present case would be zero. That 
(tan 0+ tan + tan v-+tan (2”)+....+tan(mn)) +m, 
where mn=2 7, approximates without limit to zero as 2 diminishes without 
limit, may be shown by causing z to diminish in such a manner as never to 
be an aliquot part of = or =. This may be effected by making x an odd 
194 REPORT—1857. 
aliquot part (as small as we please) of r. It is then certain that the terms 
of the above series cancel each other, and we are not embarrassed with the 
difficulty of proving that tan 5 t tan 370; for we never fall on these va- 
“a 
lues. Moreover, it has already appeared that the angle whose tangent is 
VW —] is OW —1; which corresponds with the value derived from the ex- 
pansion of tan~!@ when 2 is made equal to YW —1; while the equation 
tan—-! / —1=CC would be at variance with this expression. 
In conjunction with tan (OW —1)=W—1 it may be useful to notice, 
ay at (> 
cos (aV —1)=5 (e*+e-*); cos (OX —1l)=X; 
a itorho ve : 
sin (e@¥ —1)= , (e*—e-*) ; sin(a@V—1N=KY—1; 
and to compare these expressions with the hyperbolic tangent, cosine, and 
sine of a real angle which are respectively 1, OC, and &. 
We have already remarked that the results here contemplated, whether 
arrived at by the principle of means, or by reference to the problems producing 
them, are to be regarded not as unique values, but as interpretations guoad 
the particular subject in hand. It is not true that 1—1+1—1+.....equals © 
3 generally, or that this series has any unique value; for it may be made to 
represent any proper fraction = by making it the limit of the series 
n 
1—2 
1—an 
If A and = play, and win alternately, A winning the first, the value of A’s 
winnings is 4; but that is only on the assumption that there are no drawn 
games ; for if there be m—1 drawn games sae each game won by B before 
1 —a™ + ge? — pm tn 72n — ymt2n + y3n —_ gmt3n, +20nOL 
A wins again, the value of A’s winnings is es being the limit of the above | 
n 
series ; and if the number of games played be indefinite, the doctrine of means 
leads to this identical result. ; 
It has been suggested by Mr. De Morgan in the memoir above referred to, 
that the fabric of periodic series and integrals raised by Fourier, Poisson, 
Cauchy and others would be exposed to great danger by the production of 
any case in which |—1+....should differ from 4 when it is the limit of a 
series Aj—A,+..... If this suggestion should prove to be well-founded, it 
would lead to great doubt as to the truth of the results obtained by these 
analysts; for although I am not able to adduce any instance in which the 
known nual oem envelopment of such a seriesas #®) — xP!) + @F(2)—aO3) 4+ ..4 
differs from 4 when 2=1, yet it is easy to adduce cases in which the doctrine 
of mean values applied to such a series fails to produce + as the limiting a 
This doctrine gives as the limiting value, 
