cot (a + e) = (1 - e tan a) / (c + tan a) = [1 - e(tan a + cot a)] /tan a , 

 where a is not small. 



Estimate of Submarine Position . 



The method used to develop the approximation to the probability being 

 sought consists in first approximating X and Y by linear functions of £i 

 and JI2. The derivation of these linear functions (justified by asstunptions 

 7 and 8) is given in detail 



First, consider the linear expression for Y. Examination shows that 

 (Y/D)"-'- approximately equals 



[1 - £.i(tan a^ + cot ai)]/tan a^ - [1 - JI2 (tan a2 + cot a2)]/tan 02 



tan a2 - tan 

 tan «! tan a2 



+ Z2 



ai I 1 _ (, [ "(tan g] + cot ai)tan a2" [ 



2 j L tan a2 - tan a^ J 



[ (tan g? + cot a?) tan ai" | I 



tan g2 - tan g^ J J 



Thus , Y approximately equals 



D tan ai tan 

 tan go- tan g 



^ tan g2 | ■■ . n r(tan gi + cot gi)tan g2" | 

 - tan gj I ^L tan 02 - tan gi J 



[ "(tan g2 + cot g2)tan gi "| 

 L tan g2 - tan gi J 



Hence, Y is approximately expressed in the form a' + b'iii + CH2 , where 

 a', b' , and c' can be expressed as functions of A, B, D, on the basis of 

 the derived relation. For example it is easily verified that a' = B . 



Finally, consider the approximate linear expression for X. This can 

 be developed through multiplication of the approximate expression for Y 

 by the approximating linear expression (linear in li) for cot(g]^ + ii) . 

 This yields 



D tan g2 



tan g2 - tan 



\l + a, r 1 + (tan gi)^ ] _ j, [(tan g? + cot g2)tan gi] j 



g^ j ^|_tan g2 - tan giJ ^L tan g2 - tan gi J j 



as the approximate expression for X. Thus, X is expressed in the form 

 a + bJ!-! + c£2 t where it can be shown that a = A . Also b and c can be 

 expressed as functions of A, B, D, that are determined by the derived 

 expression. 



Relation for Inclusion in Circle . 



It is easily verified that the submarine location is contained in the 

 circle of radius R centered at (X,Y) if and only if 



(X - A)2 + (Y - B)2 <_ r2 . 

 452 



