square of the mean error, e, or of the "probable error" />, 



iv x l/f« x If? etc. 



The following fundamental equations may be derived 



from the theorem that to form the sum of quantities for 



which the signs + or — are equally probable, we must 



take the lines as vectors at right angles to each other. 



Let the corresponding Greek letters «, /?, y, etc. denote 

 the mean (or the probable) error of the quantities a, b, c, 

 etc., then we shall have 



(a ± a) + { b±fl) = (a + b))l±,'(<c + jJ 2 )/(a + b)[ (35) 



(a ± a) x (b ± P) = ( ,b>l ±4(«M 2 + (P/bY)} (36) 



(a * «) + (b * p) =|{l ± 4(a/a> + (/3/b)'] } (37) 



Let >/, with a corresponding sullix, denote tiie mean (or 

 the probable) error of any ordinate ;/. Then from (29a) we 

 have, for the errors of the numerator and demoniinator, 

 which latter for brevity may be respectively denoted by V 

 ami r, the value ..f r»0a - fiillv evpr.-ss.-d, viz.. 



curve may be set out as follows :— 



of the probabdity 



In terms of the symbol Mod « lM J^S« 



" Mean | Probable 



Modulus k [ 1-000 ! 0-564 

 Mean of errors m \ 1-772 1-000 

 "Mean error" e 1 1-4H o-7'...s 

 Probable error ,, -2-007 1-183 



I 0-707 0-477 



1-253 0-845 



1 1-000 1 o' 674 



i 1-483 ] 1-000 



A fundamental theorem coinie,-,:. i 





