141 



IX. — Distance of objects at sea. 



[April 



To the Editor of the Madras Journal 



of Literature and Science, 



Sir, 



In Adam's Geometrical and Gra- 

 phical Essays at page 117, the follow- 

 ing problem is given, viz. — The dis* 

 tance of three objects A, B & C, from 

 each other, and the angles ADC, 

 C D E, C E D, C E B being given, 

 to find the sides A D, D C, D E, 

 E C and E B. 



The method of construction is there 

 given, and also an indirect method of 

 calculation, but a direct general for- 

 mula for the solution of the problems 

 may be found. To simplify the cal- 

 culation, let the three points A C B 

 be conceived in a straight line, and 

 producing the lines B E and A D to 

 M. 



a -f £ -f S -f € — 180°, and calling the angles 

 x then 



(a -f /3 -f- 5 f e)then 



the angle M 

 ABE = y and BAD 



x + y= 180° — M = 360° 



in the triangle C B E we have CE =CBx~^ 

 and in the triangle D C E we have 



jy C E. sin (e -f c) C E. sin y s*n (e -f- S) 



sin 6 sin ft sin e 



In the triangle C A D we have DC= : * 

 and in the triangle D C E we have 



E D 



D C. sin (e -f- AC. sin x sin ( e -f- ®) 



sin g 



and equating the two values of ED there results 



C B. sin a sin £ sin x 



A C, sin ftnin c 1 sin y 



whence 



