96 EIGHTH REPORT 1838. 



„ -4549 S (* - aY 



(12) 



n-1 



so that e* = « E*. The weights, in both cases, are measured by 

 the inverse of the squares of the probable errors ; that is 



WE2 = ], we^= 1, (13) 



tv and W denoting the weights of the single result, and of the 

 mean, respectively*. 



When the quantity sought is a linear function of two or 

 more unknown quantities, which latter are obtained imme- 

 diately by observation, its probable error is connected with 

 those of the quantities on which it depends by a very simple 

 relation. 



Let X and y be the quantities sought by immediate observa- 

 tion, and let the quantity actually sought, ^, be a linear func- 

 tion of these, expressed by the equation 



z=px + qy. 



Let a denote the ai-ithmetical mean of m observations of the 

 unknown quantity x \ h the mean of n observations of y ; and 

 let E and E be their probable errors, or the limits on either 



side of which there ai'e equal chances o{ the actual errors, x—a, 

 y — b, being found. Then the probable error of .?, E^, is ex- 

 pressed by the formula f 



E^=^^E^ + ^,E^. (14) 



The case of a linear function includes every case in which 

 the quantities sought are already approximately known. We 

 have only to substitute for these quantities their approximate 

 y?L\\ies plus the unknown corrections, and to neglect the squares 

 and higher powers of the latter. 



To apply these principles to an important case, — let it be re- 

 quired to determine the probable error (or the weight) of the 

 mean dip at a given station, as deduced from n^ observations, 

 with n. instruments. 



The true dip being equal to the observed dip p/M« the in- 

 strumental correction, it is manifest that, in this case, 



E^ = E^„+EV 



* For the demonstration of these theorems, the reader is referred to a paper 

 by Prof. Encke, in the Aslronomisches Jahrhuch for the year 1834. See also a 

 paper by M. Poisson on the same subject in the Connohsance des Temps, 1827. 



I See a paper by M. Poisson in the Btdletin Universel des Sciences, tome xiii, 

 p. 266, See also the Memoir by Prof. Encke, already referred to. 



