264 Theory of Transit Corrections. 



Differentiating and establishing the condition of a maximum, 



sin 8" 

 we find sec 8" — c' = (tan $" . cos <p - c") . • Substituting in 



cos (f 



sin #" 



the original expression, it reduces to ; the greatest possible 



cos w 



value of which in our lat. is 1-33. Taking this twice we have 

 2*6. Making, thenm — m' very large in expression (2), let us 

 ascertain its greatest value. It is evident that this expression is 



m" — ra' 



the greatest, other things being equal, when — r, -7 is the great- 



n ' — n 



est, i. e. when 8" is the greatest and 8 / is the least. Taking 



n" — n f 



<5"=88° 30'=declin. of Polaris and <S'=0, we find -- ; =-1*29. 



Now the maximum value of expression (2) is when sin &' 



n" — n' 



-—77 — —, . cos y= — , 97 in 



: 



5"=76° 56', and may be found by calculation to be 76. Hence 

 it appears that if one of the two coefficients is a small fraction 

 the other must be integral. 



The remaining point is evidently established, if we show that 

 there is no likelihood that either set of errors is greater than the 

 corresponding single error of the other expression. Each set of 

 errors in the expression which gives the effect of the errors of 

 observation in the common method, is the algebraic sum of a 

 number of errors. Let us conceive two of these errors to be 

 taken at hazard. Then there is no probability that their algebraic 

 sum is greater than the greatest of them, inasmuch as there is no 

 probability that both have the same sign. Now let another error 

 be taken in the same manner. Then there is no probability that 

 the sum of the three is greater than the sum of the two first, for 

 the same reason as before : and, as there is no likelihood that the 

 sum of the two is greater than the greatest of them, there is 

 none that the sum of the three is greater than the greatest of the 

 two errors taken at hazard. In the same manner we might go 

 on adding single errors and reasoning reversely, and thus prove 

 that there is no probability that the algebraic sum of any gi ven 

 number of errors is greater than the greatest of any two of them, 

 taken at random. But now there is no probability that this 

 greatest of the two is greater than the error corresponding to the 

 set in the first factor of the second expression, and therefore none 

 that the algebraic sum of all the errors constituting the set, is 

 greater than this error. 



