190 REPORT—1857. 
almost irrational idea, of the amount of longitude which it would be neces- 
sary to traverse before we reach the pole, the condition of the problem being 
that we are never to quit the equator. It is useful, however, not to neglect 
this transitional infinity, for it is evidently of a different kind from those on 
each side of it. It is the infinitely remote limit of a series of tnmereasing in- 
finities, just as the other transitional point appeared to be the nearest approach 
to finite quantity that infinity admits of, the minimum point of a series of 
decreasing infinities. 
Before proceeding to consider the trigonometrical functions of Oz, it will be 
convenient to point out another series of infinite angles which presents itself 
in the course of this investigation, and which may perhaps throw some light 
on the nature of an infinite angle, though it is not so readily susceptible ef 
geometrical illustration as the case already considered. It has been observed 
that any angular position furnishes an infinite number of values of 6,, and 
that the longitude has a corresponding number of values, connected by the 
equation 
(4+2n 7) tan 6n=log cotan 5 
By increasing (which denotes the number of circuits which the course 
makes round the sphere) without limit, 6,, diminishes without limit; and the 
course approximates to an infinite number of circuits round the equator. 
The series of infinite longitudes now intended to be brought under considera- 
tion, consists of the limits of the values of the longitude traversed when 6, ap- 
proaches 0, for all positions on the sphere from the equator to the pole. 
When the point is at the pole, the infinite angle under consideration is that 
which we have already noticed as being of a very high order of magnitude ; 
its value is infinite, not merely because tan On is zero, a circumstance which 
for our present purpose is common to all other positions, but also because 
log cotan 5 being log tan 5 is infinite. As the position descends from the 
pole towards the equator, the limit of the longitude traversed assumes still an 
infinite value, but diminishes in proportion to log cotan = until the point 
falls upon the equator itself, in which case the value is a transitional phase of 
the series of infinite angles, and it assumes the form 0 +0, where the two ele- 
ments of which the zeros are the limits are independent of each other; from 
which we are to infer, what is indeed geometrically evident, that the course 
consists of an arbitrary number of complete revolutions 2 ” 7, in addition to 
the original longitude of the point Z.. On the other side of the equator, the 
longitude traversed passes through an exactly similar series of values, but 
without any change of sign. We have thus another point of view from which 
we perceive that the limiting conception of a series of infinite angles of vary- 
ing magnitudes proves to be an angle possessing an infinite number of values. 
Having thus acquired some idea of the meaning of an infinite angle, by ob- 
serving its demeanour through the various phases of the above geometrical 
illustrations, the next question which suggests itself is, how are we to deal 
with its trigonometrical functions. Confining ourselves to the set of angles 
which we have denoted by QC, (the longitude traversed in reaching the pole 
at the angle 0), we may safely assert that these functions are independent of 
6; for whatever may be the value of 6, all the values of OC, corresponding to 
them are geometrically identical; and we are here dealing with purely geo- 
metrical functions. It follows that a trigonometrical function of an infinite 
