ON THE ALGEBRAIC COUPLE. 187 
after the fashion of Mercator’s chart. The product of two positions (J, 0) 
and (J! 6'), (that is, the fourth proportional to the unit of angular magnitude 
and the two positions, ) is evidently the position (2, 6"), where 6 =6' + 6, and 
it 
pretest ; that is to say, the amplitude of the product is the sum 
relation to plane angular magnitude. 
It will be at once apparent that in this method of representing angular 
position, every point has an infinite number of symbolical representations, 
inasmuch as it may be reached from the origin by an infinite number of per- 
fectly distinct courses. ‘The longitude which we have denoted by / may be 
denoted also by any value of 2+2 7; each value of which has a correspond- 
ing and distinct value of 6. In effect, the couple becomes 
z sir) nw 9 nvm] 
cos 0, ‘ 
where 9, is determined by the equation 
(442n7) tan 0,=log cot 5 
If the point be at the pole of the sphere, the values of 0, are continuous ; 
that is to say, every angle is a value of 0,; which is well known, for it is ob- 
vious that the pole may be reached by an unvarying course at whatever angle 
we start from the equator. In this case the longitude traversed is infinite ; 
except when 86, is 7 in which case the longitude traversed assumes the form 
OC+oc, and is any arbitrary quantity. This also is geometrically evident, 
since if the pole be regarded as having a longitude, that longitude is per- 
fectly arbitrary. In this case the couple assumes the form (a + /—1 tan 5). 
_ Comparing this with the form we started from, tan—} Yt tan-1@V 1 
3 and remembering that at the pole z=7, we have tan—! ( V—-1)=av —1,; 
_ the infinity employed being an arbitrary multiple of tan 5 
Since 
a+ VV hy 
. 1—aV—1 
it follows that, if either of two positions be at the pole, their sum is at the 
_ pole; unless indeed the other point be at the opposite pole, (in which case 
a — / —1,) when their sum is any point on the equator, or rather every 
_ point on the equator at the same time. In like manner, since 
tan-!¢—tan-! (/—1)=tan-! se Miob her ony /—1), 
1+a 1 
tan—'a+tan-! (W—1)=tan-! =tan-14/ =], 
__ it follows that if a position at the pole be deducted from any other position, 
_ the resultant position is at the opposite pole; and since 2 tan—} (Vv —1) 
eet tY—D, the sum of any number of polar positions is a polar posi- 
ne 
