Heights of the principal Hills of Dent, fyc. Yorkshire. 89 



Consequently when either a or d are obtuse, the mean error 

 of calculation will be 



a! — — , when a 1 is greater than d! \ 



— 7—5 when d! is greater than twice a 1 ; 



—p — , when d! is greater than a! but less than ZaK 



x as thus expressed is the logarithm of the natural number 

 by which we must multiply the correct distance, in order to 

 obtain its value as affected by the mean of the possible errors 

 of observation arranged conformable to our list. Consequently 

 if we call 1 the correct distance, then will the error of mea- 

 surement be equal to this natural number minus 1. Now if 

 the distance as derived from one base A shall be liable to an 

 error of 10 feet, and the same distance, as deduced from an- 

 other base B, be uncertain to 20 feet, then must the claim to 

 accuracy of A exceed that of B in the reciprocal ratio of 10 

 to 20 or as 10 to 5. 



When we have given P, F", P" or the required distance as 

 calculated from the same number of bases, together with the 

 reciprocals of x — 1 corresponding to the respective bases, 

 which call tf, x", •*•"', the correct value of F may be considered 

 as equal to 



x 4- x" + *'" ' 



In practice it will be most convenient to take out the log. dif- 

 ference for 1' of the sines of the angles d and e, noting them 

 + or — according as the angle is acute or obtuse. A type 

 of calculation is subjoined. 



Base A; Noughtberry Hill to Ingleborough . . 44106 feet, 

 B Ditto Dod Fell .... 19347 



A. Noughtberry 62° 19' 8" 



Ingleborough 55 26 53 log. diff. of 1'= 4- 0-0000870=*/', 

 Berkin ( 62 13 59) 4- 0*0000665 = a!. 



B. Noughtberry 145° 40' 35" 



Dod Fell 23 30 40 log.diff.of l'=e 4-0-0002904=^, 

 Berkin (10 48 45) ...... 4-0*0006610=0'. 



Berkin to Noughtberry Hill by base A . . 41053 feet. 

 Ditto Ditto B . . 41142 



Arithmetical mean 41097*5 



x for the base A will be a' + — , =0*00 

 New Series. Vol. 3. No. 14. Feb. 1828. N of 



The value of x for the base A will be a' 4- -^, =0*0001 317, 



