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of ori'oi'S and to a i'oniail<a!)lo aiialoüv I)ot\vooii lliis prohlrMU and 

 that of llio iHoiiKMits of iiiorlia in dvnaniics. 



If wc take A\ tli(^ nnrnhei' of observalions. equal lo unity, llie 

 relative frequency of the ends or representative |)oinls of the veetoi's 

 may be represented l)y ilie density of these points per unity of 

 surface. This function of ju-obabilily is called by Schols "the module", 

 the "specific probability" oi- Ihe "facility de I'erreur". 



We thus obtain a mechanical ima,i>e of a surface of probabihtv, 

 the density of wliicli will, in lienei-al, be a function of the length 

 and direction of tlie vectors. 



The determination of what Schot,s calls the constant pai't of the 

 error — the prol)ability of which is X=ï — is identical with the 

 determination of the situatioji of the centre of gravity, and the 

 calculation of the mean (not average) error: 



M = 



\/'^ 



with that of the moment of inertia, Avhicli leads to the determination 

 of two (in the plane) principal axes of inertia, which, in our case, 

 may be called axes of probability. 



Assuming that these errors in the plane are due to the coo|)eration 

 of a great number of elementary eiTors, SrnoLS has proved that the 

 projections of the errors on an arlutrary axis follow the exj)onential 

 law of errors in a line and that the law of the resulting eiTor can 

 be found by supposing the error to originate in the coincidence of 

 projections of the eri-or n|)on the axes of probability, these j)rojec- 

 tions being regarded as independent of each other. 



The application of this theory to our case can be reduced to very 

 simple calculations. 



Errors arising from ilHli^•idnal or instrnmeiital canses are always 

 distributed in a more or less systematical way, but thei'c is no 

 reason lo suppose that the fluctuations e.g. of barometric heights 

 within an arbiti-ary length of time, and cleared from their I'onslant 

 part, ^vill show any teiuleucy to systematic distribution when arrajiged 

 around a |K)int in the way described above. 



ScHOLs' specific j)robability of an error in the plane is given by 

 the expression: 





in which x and y are the coordinates of the error (polar coord. 9 and 6) 



