In October 1913, Chatfield (Beatty’s flag captain) wrote a memorandum entitled “Fast Division Work from a Gunnery Standpoint” (The Beatty Papers Vol I, Item 49, page 90). This memorandum included the following statements about changes in gunnery range:
“… the attempt to obtain a tactical, or rather a gunnery advantage, usually results in a high and frequently changing rate due to constant change of course. This must affect the gun fire, possibly … to such an extent as to entirely neutralize the value of the position gained.”
“… it is quite easy, with superior speed, to calculate suitable courses which will keep the range constant and the rate nil …”
“The T must never be crossed at too broad an angle as this is unnecessary and causes a big and difficult rate.”
Two examples are given in the memorandum for a fast division speed 5 knots greater than the enemy. Both are apparently wrong and were corrected in notes added by Beatty. Also, specifying only the difference in speed is insufficient. The solution for 10 knots and 15 knots (for example) would be significantly different from the solution for 20 knots and 25 knots.
The following diagram shows the problem in a general form, where:
Ao Angle before the beam of the enemy
De Distance traveled by the enemy unit over the time interval
Df Distance traveled by your unit over the time interval
R Range to the enemy unit
A Angle to steer toward enemy
Although Chatfield says he can calculate ‘perfect gunnery courses’, the range is not precisely constant over the time interval. It is only the same at the start and end points. Line Df would need to be a curve (implying a continuous change of course) to keep the range constant at all times.
A general solution to the problem can be developed by using the Law of Cosines. The area swept by the fire range is divided into two triangles:
To keep the formulas to a manageable size, intermediate terms are calculated:
Tables for various combinations of speeds, ranges and angles can be created: