ei : 
On a plane the direction of unvarying course is the straight line; and its 
equation is 
186 REPORT—1857. 
y=a tan p, 
and the symbol denoting the point (2, y), involving its distance along the 
unit-line, z, and the angle defining its course, , is . 
71+ V—] tan 8); or 4 ebV-1, 
Now if, on a sphere, we consider the equatof as the unit-line on which real 
angles are measured, the line of unvarying course, or the line which is always 
inclined to the unit-line at the same angle, is the rhumb line; and its equa- 
tion is 
Ztan @=log cot a 
where 6 is the angle of direction. Introducing this value of 6 into the couple 
in its new form, J+ /—1 log cot we obtain simply 
1+ 7 —1 tan 6), or lea 
cos 6 
which is a couple of precisely the same form as the ordinary plane couple. 
If we lay down the definitions that two lines of unvarying course (or 
rhumb lines) are parallel when they are inclined at the same angle to the 
equator, and that parallel lines are equal when they traverse the same amount 
of longitude, we easily obtain the proper rules for adding and multiplying 
two or more angular positions. 
To find the sum of two positions P or (/,@) and P’ or (7, 6'). From the 
origin O draw the unvarying courses OP and OP’; from P draw the course 
PP" equal and parallel to OP’; then P" is the resultant position. For the 
algebraic sum, 
11+ V—1 tan 0) +2 (1+ ¥ —i1 tan 6’) equals 2” (1+ / —1 tan 6"), 
if 24-U=l"", and é" tan 6’=7 tan 0+7' tan 6. 
Now in the construction above given, (which the reader will easily imagine 
without diagram,) we have ; 
Mai+0; 
and since the equation of the line PP" is 
" 
' f— fete B 
l’ tan 6'=log cot 5 log cot 3 
we also have 
l" tan 6"=/ tan 6+/7' tan 6’. 
Similarly, to find the difference of two positions P! and P”: from P! draw 
the course PP parallel to OP! and equal to it, but in the opposite direction ; 
then P will be the difference of the two positions. 
Considering the couple in the form a ile it will be seen that the 
amplitude denotes, as on a plane, the direction of the position, or the angle 
defining the course ; but the modulus does not denote the angular distance — 
traversed, or the length of the course, for that is o 53 but it denotes the 
sin 
line into which the course would be projected if the sphere were opened out 
