LITTLEHALES: THE SUMNER LINE 507 



From which it appears that as long as the product of cot h cot Z 

 is a proper fraction, the uncertainty in the calculated value of h 

 will never exceed the uncertainty in the value of Z, that there 

 will be no uncertainty in the calculated value of h when the ob- 

 served body is on the prime vertical and also when it is in the 

 zenith, and that there will be a maxhnmn uncertainty when the 

 observed body is on the meridian. 



The curve of maximmn numerical uncertainty in the cal- 

 culated value of h, derived from the second differential equa- 

 tion of h with respect to Z, may be drawn for any given lati- 

 tude, L, from the equation: 



cos^ Z — cos^ h cos Z — sin h cos h tan L = 



For purposes of illustration, the solution of the example, stated 

 as follows under Article 372 of the American Practical Navigator, 

 is given: 



At sea May 18, 1915, A.M., in latitude 41° 33' N., longitude 33° 30' 

 W., by dead reckoning, the mean of a series of observed altitudes of 

 the sun's lower limb was 29° 41' 00"; the mean watch time, 7*^20™ 

 45.3^*='=; C.C, + 4-59.2-- I.C., - 0' 30"; height of eye 23 feet; 

 C. — W., 2'> 17'° 06-". Required the Sumner line. 

 Mean W. T. 7'^20'M5.3^'* Eq. T. 3'M4.48^ Dec 19°22'27.9"N 



+ 33.65" 



G.M.T 

 corr 



C.C. +4 59.2 Eq.T. 3 - 44.6 Dec. 1 9-21 UN 



G.M.T. 17^ 21 42 50.5 (Plus to mean time) p 70 38 49 



Eq.T. +3 44. 6 

 G.A.T. 21 46 35.1 



Lon.by DR 33° 30' 2 14 00 

 L.A.T. 19 32 35.1 



Under the principles laid down, it now becomes the object to 

 assume a geographical position in the nearest longitude to that 

 given by the dead reckoning which, when applied to the Green- 



