204 EEroET— 1882. 



These are, of course, not comparable with the others above obtained, as 

 the subdivision of the tube is different. 



If the tube is divided into five jDarts by Thiesen's method the (p. e)^ 

 = 0-32e2. 



Bessel's Method. 



The above method is not strictly applicable to Bessel's corrections, 

 but it may be approximately applied as follows : — 



Each value of h,. depends on 10 independent measurements. Hence 



{p. e. h,y = — , &c. 



Each value of ^ (Uir) depends on 100 measures. 



Ten of these arc introduced with the coefficient O'l in r^., ten (of 

 which one is common to this and last group) with the coefficient O'l in 

 h^, and 100 with the coefficient — 0*01 in the m. One, viz., U/^^ itself, 

 appears again with the coefficient — 1. Hence 81 will have the 

 coefficient — 001, 18 will have the coefficient 0'09, and 1 the coefficient 

 — 1 + 0"2 — 001 ^ — '81. Hence the probable error of a single thread- 

 correction is : — 



{81 X 0-0001 + 18 X 0-0081 + (-81) '-} e^ = -81 e\ 



a result which strongly confirms the view expressed above as to the rela- 

 tive values of the initial point and other corrections. 



Let it next be supposed that all the corrections in the diagonal which 

 passes from the lower left to the upper right-hand corner of Table XIX. 

 (p. 180), refer to the same point. 



Hence the final correction will be the mean of the ten. These are not 

 independent. Each number in the Table will occur ten times. 



Ten, viz., the numbers in the diagonal, will have the coefficient — 081 

 once, and — O'Ol nine times. Their coefficients in the mean will therefore 

 be — 0-09. The remaining ninety will have the coefficient 0-09 twice, 

 and — 0-01 eight times. Their coefficients in the mean will therefore be 



o 



^ I 0-18 -0-08 I =0-01 



Hence the probable error of the correction will be e^, multiplied by 



10 X (009)2 + 90 X (0-01)2 ^ 0-081 + 0-009 = 0-09. 



For comparison with the other methods referred to, it would be better 

 to take Bessel's method with five principal points 4° apart. The probable 

 errors of an initial point, and a mean upper point, would then be 0-2 e^ and 

 *15 e^, respectively. 



