280 PROFESSOR K. PEARSON AND MR. L. N. G. F1LON 



and the correlations 



P B " P _A _ P B 



KM v/(BuB.) ' """ " v/(B n B S! ) ' - vABuB,,) . 



T r IL. f ' ' ' (cvi ' ) - 



-p _ PM -p _ __ 31 _ p __ 5* 



Lt( "'" - v/(B K B Si ) ' - y(B w B u ) ' *"*" - v/CBsA.) J 



The algebraic expressions for the expanded determinant and its minors are very 

 lengthy, and it will be found easiest in any numerical case to calculate the numerical 

 values of the b n> l>\-i, ^i a , b lt> b-n , and then find the values of the determinant and 

 its minors numerically. So soon as the above four standard deviations and six 

 error correlations have been calculated, the determination of the probable errors in 

 the fundamental constants of the frequency distribution becomes easy. 



We have for the mean organ M ( , if Ji now defines the origin of coordinates, 



AM t = A/i, + Ax 



. ///,+ ! I (//*,+ 1) Aw, li(nt, + 1) AM. - .. , 



Whence 



, + 1 r v ., 4 s (HI, + I)'-' , ir (w, + 1)- 



H- -j- a^-h - 



., 





1 )(>, + I) ^(/w, +_!).-- p 



^ -2 6 S (lil K Swi - -jj- SftS^Ki,,,,, 



!)(M, + 1) 



, - - S^B ....... , ........... (cvm.). 



Similarly, the modal value of the organ can be found from* 



in. l)n 6w, . , 



AM = A/i + - A?> + . AJH, ,., A?H, . (cix.). 



//., + m., (Ill, + M.,)* (7/lj + ?H S )- 



* The easiest numerical method in this, as in the previous case, -is to proceed as follows : Write 



v _ v . 111, v , &TOJ bm { v 



' Mh + 7/1, + in, "^^ (m, + m,) 2 " l ' Xn " ~ (in, -f- ^ i ^i 1 



where the x' *"'e umbral symbols, and let N,, N. ; . N 3 , N 4 be the numerical values of the coefficients on 

 the right. Then put 



-M X Mo = N lX/l + N 2 , V/J + N,X- WI - N lXw/ 



Now square this equation, and, whenever a product, x 7 X,/ occurs, multiply it by the corresponding error 

 correlation, R, /v ', already calculated, putting R / = 1 if 5 = 5'. Then, actually, 



SM O = N? + NJ + NJ + Nj + aN.N.Rift + 2N 1 N 8 R/,,, ll - '2N.N.R/,,,,,, + 2N s N s R 6mi 



BARLOW'S Tables rapidly give the squares and CRELLE'S Tables, or a Brunsviga, the products. 



