ON THE ALGEBRAIC COUPLE. 189 
a fixed origin has this singular value. If the conception of such an angle be 
difficult, the difficulty exists to the analyst alone: to the geometrician the 
idea is easy and even elementary; for to assert that the pole has at once 
every possible longitude, is merely to say that it is the intersection of all the 
meridians, or that it exists at once on every meridian, which is in truth the 
definition of the pole. . 
The considerations here developed seem to derive additional weight from 
a view of the subject which we have hitherto excluded; where the point 
passes to the pole directly in latitude, without any geometrical change of 
longitude ; that is, the case where a=— This is a limiting case. For all 
values of 6. up to this value, the longitude traversed is infinite ; a¢ this point 
the infinity changes sign. What is the value through which it passes? The 
analytical expression for the longitude, in all cases, is 
l=cot 6 log cot Bats log tan ; at the pole. 
This then assumes the form QC+QC, where the elements producing the two 
infinities are independent of each other ; a kind of expression which we know 
to be the algebraic symbol for that which has every value within the whole 
range of value. It is indeed generally said to be the representative of inde- 
terminate value, or that it means, “any real quantity we please:” but the 
language appears inadequate, the thing represented being manifestly, “ every 
real quantity at once.” Here then we have a case in which, without passing 
from the field of analysis, we find the conception of an angle having every 
possible value at once ; and it presents itself to us as the limiting idea of a 
series of infinities, and as the mode of transition from a series of positive in- 
finities to exactly the same series of negative infinities. 
The series of infinite angles which represents the longitude traversed in 
reaching the pole, is a series of functions of 0, of such anature that their rate 
of increase can be ascertained and their relative magnitudes compared, with 
as much ease as if they were all finite quantities ; for they vary directly as 
cotan 0. 
If OC, denote the longitude of the pole as reached at the angle 0, we have 
Bb 
i is eb tan de" 2 Oey: 
do sin?@ sin? 0 sin20 
an equation which marks the rate of decrease as 0 advances from 0 towards 
ae at that point it passes through the phase of analytical indeterminateness, 
and then passes through the stages of negative infinity at the same rate of 
progress as it had manifested on the positive side. The general result may 
be thus recapitulated : that having in analysis met with angles of various de- 
grees of infinite magnitude, they are interpreted geometrically into angles, 
each of which has at the same time every possible value ; that having also in 
analysis met with an angle which has an absolutely indeterminate value, or all 
values at the same time, we find that it is the mode by which a series of de- 
creasing positive infinite angles passes without discontinuity to an exactly 
similar series of negative infinite angles. 
_ The series of infinite angles with which we are dealing is evidently at its 
positive maximum when 6=0, and at its negative maximum when @=z. 
At these points we pass through what would appear to be the most trans- 
cendental of all infinite angles; which resolves itself in geometry into the 
