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These are the special cases and rules to deal with the special cases used on my version Flat Bygrave. These special cases relate only to Bygrade slide rule computations due to limitations of the slide rule scales, they do not apply to using the Bygrave formulas on a calculator or when using logrithmic trig tables. 

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SPECIAL RULES


There are some unusual cases that require slightly different procedures and all of these special cases are described on the form. If "H" is less than 1º or greater than 89º simply assume a longitude to bring "H" within the range of the scales. The intercept will be longer but perfectly usable for practical navigation.

If the computed azimuth angle is greater than 85º the computed altitude will lose accuracy even though the azimuth is accurate because with azimuth angles in this range, even rounding the azimuth angle up or down one half minute can change the Hc by ten minutes. So you use the azimuth you have calculated but you compute altitude by interchanging declination and latitude and then doing the normal computation a second time. You discard the azimuth derived during this second computation of altitude and use the original azimuth.

When declination is less than one degree you can't begin the computation the normal way to find "W" because you have to start the process with declination on the cotangent scale and this scale doesn't extend below 1º. So in this case you just skip the computation of "W" and simply set "W" equal to declination. Using this method you arrive at an azimuth that is not exact but is a close approximation and in the worst case I have found the azimuth is still within 0.9º of the true azimuth but most are much closer. If the declination is less than one degree and the latitude is also less than one degree, follow this procedure and also assume a latitude equal to one degree. After you have computed the Az you then follow the same procedure discussed above for azimuths exceeding 85º by interchanging the latitude and declination and then computing Hc which will produce an exact value of Hc.

Another rare possibility is that "Y" will be less than one degree or that it will exceed 89º after adding "W" to co-declination so it won't fit on the scale. The simple way to handle this situation is to assume a different latitude so the "Y" does fit on the scale even though the resulting intercept is longer but still usable.

An extremely unlikely case (I only mention it to be complete) is that "W" exceeds the range of the cotangent scale, 89º15', so cannot be computed in the first step of the process.  This can only happen when shooting one star, Kochab, which has a declination of 74º 07' north and then only if "H" exceeds 87º 20', an extremely unlikely event.

The attached form steps you through the computation and contains the rules for the special cases. The special cases are likely to come up only very rarely in practice.

The first rule for H < 1º or H > 89º only involves LHAs covering 5 degrees out of 360º (LHA in the ranges of 0 -1, 89-91, 269-271, and 359-360) so only occurs by chance very rarely and these can be avoided if sights are preplanned as is the normal procedure for flight navigation. In the worst case, you have to change the time of the observation by four minutes.

Rule 3 covers the case when Y is less than 1 or when Y exceeds 89º which covers a range of three degrees out of a possible 180º so is also very rare. Co-lat is in the range of 0º-90º and W is also in the same range so X comes in the range of 0º -180º. If X is less than 89º then Y is also less than 89º. If X is greater than 91º then Y is also less than 89º. Only in the case of X between 89º and 91º will Y exceed 89º. This situation can't be avoided in advance because you can't predict what the value of W will be but if it occurs then just assuming a latitude that differs at most by one degree solves the problem which will result in a longer intercept but one that is still usable.

The fourth rule deals with cases of bodies bearing almost directly east or west and this situation can be avoided by choosing a different body to shoot or, if only the sun is available, by waiting a few minutes to allow the azimuth to change out of this range.

The remaining situation covered by rule two (declinations less than one degree) concerns only bodies in the solar system since none of the navigational stars have declinations less than one degree. Obviously the most important body is the sun and its declination is between 1º north and 1º south for five days in March and again in September so this situation can't be avoided and this is the most important special case. The special rule handles it nicely and the Hc is completely accurate. The computed azimuth is an approximation but is never more than one degree different than the actual azimuth and is usually much closer. Since you can use your D.R. for the A.P. for this situation the intercepts can be kept short and the slight inaccuracy in the azimuth will not make a noticeable difference in the LOP.


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EXPLANATION OF AZIMUTH RULES
 

In most situations there is no ambiguity as to which 
quarter the Zn lies since you know the approximate direction you are 
looking when you take the sight. The problem arises because the azimuth 
angle is limited to the range of zero to 90 degrees and when the Zn is near 
east or west the correct Zn might fall either 
side of the line so there is an ambiguity in converting from azimuth 
angle to Zn.

One easy rule to apply first is that if the declination is greater than 
the latitude then the azimuth can never be in the opposite semicircle. 
To generalize this rule, if the declination has 
the same name as the latitude and the declination is greater than the 
latitude, then you start with the direction of the elevated pole (the 
nearer pole) when converting from azimuth angle to azimuth (Zn.)

The second rule to apply is that if the declination is contrary then the Zn 
must be in the opposite semicircle. To generalize this rule, if the 
declination and the latitude have contrary names then you start with the 
direction of the depressed pole (the further pole) when converting from 
azimuth angle to Zn.

These two rules take care of most of the cases, especially for 
navigators in low latitudes.

The remaining ambiguity concerns situations in which the declination is 
the same name as the latitude but is less than the latitude. In this 
situation the azimuth of the body will be both north and south of the 
east - west line during part of each day. The rules compare "W" 
with latitude to resolve this remaining ambiguity.

So, combining all three rules:

If the declination or if "W" is greater than the latitude then combine 
the azimuth angle with the direction of the elevated pole.
If the declination name is contrary to the name of the latitude or if 
"W" is less than the latitude then combine azimuth angle with the 
direction of the depressed pole.

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