ON THE ALGEBRAIC COUPLE. 193 
This consideration enables us to obtain the following results, without the ap- 
plication of the principle of mean values, which, however, gives the same 
results :— , 
x _# 
Since 2 sin z= — V —1((—1)* —(—1) 7), we have sinQC=0. 
Dabs Seg A ae Ba eek 
Since tan c=— VW —1 Mone Salil) we havetano¢= — VW —1(4—})= 
(-1)"+(=1)* 
and generally, the odd functions and their odd powers will be found to be zero. 
The value of the even functions is not immediately perceived, untess we con- 
rs 
& 
ceive ourselves at liberty to assume that (—1)*™ and (—1) 7 become 0 when 
 xisinfinite. If, however, we bear in mind, that in speaking of OC as an angle, 
we are using the symbol in its ordinary algebraical sense of a number which 
_is larger than any number we can name, and which does not admit of alte- 
ration by the mere addition or subtraction of any finite quantity, and con- 
cerning whose value nothing further can be predicated, we shall easily per- 
ceive that cos OC must have the same value as sin OC; for since 
. Tv . T 
a —_)j=—s e—— 
cos v=sin (#+5) sin ( =) 
we have, with the above meaning of infinity, 
cos OC= +sin C, and therefore =0; 
from which we are to infer that (—1)%* and (—1])~%, or e#%¥=1, are both 
zero, since they are equal and their sum is zero. 
si Ad sin w A cos # 1 
Since tan piste 20 when zw is O= ““_, we have tan O0=— ——_.. 
cosz O —sin & tan OC 
By taking the powers of 2 cos e=(—1ye+ (1) *, 
and of Qing — /—1((—1)"—(—1) *), 
x é 
_and observing that (—1)" and (—1) 7 are both zero when 2=Q, we ob- 
tain 
1 2n(2n—1)(2n—2)...(m+1), 
2 1.2.3....7 : 
, cos22+19¢=0 3 
and the powers of the sines are the same. The assertion that tan OC=0 is at 
variance with the result given by Professor De Morgan in his treatise on di- 
vergent series in the Cambridge Phil. Trans. (vol. viii. part 2). The result 
_ deduced in that paper by two methods is'tan O0=-+WV—I. The first me- 
- thod depends on the assertion that log (Sav —1, and log (=) 
cos”"QC = 
=rV—1; but inasmuch as the two fractions whose logarithms are required 
are, from their derivation, reciprocals of each other, I apprehend that the sum 
_ of their logarithms is simply the logarithm of 1, or zero; which would lead 
to tan OC=0. The second method leads to the equation above deduced, 
7 from which however we are not at liberty to deduce 
tan o¢= + /—1. Independently of the @ priori difficulty of believing that 
the mean value of an odd function can be other than zero, I conceive that any 
O57. o 
