STUDIES IN STATISTICAL REPRESENTATION. 103 



the coefficients of b, c, d, e in the successive terms r u r 2 , 

 etc., being represented by the following scheme 



r b c d e r b c d e 



l+ra+i+i' + ifc 7 -m -i i +ik 

 2+Z+Z+ir 8-1 -I \ 



3 +i +m +i -\k 9 -\ -m \ -\k 



4 +i -m-i -Ifc 10 -| +m -i -ik 



5 + z - j - \ o ii ' -i +r -4 o 



6 +m -| -i +Jfc 12 -m +| -£ + ifc 



From these necessary combinations may be obtained for 

 determining the constants b, c, etc. 



The most probable values of these constants may be 

 obtained from the whole of the equations by making use of 

 the method of least squares. 



Multiplying the equations in turn by m, J, i, etc., (i.e., by 

 the different co-efficients of b) we obtain, since the sum of 

 the co-efficients of c, d, e all total to zero, 



(67) ... ^ {mn + lr 2 + \ r, + }= 6 (2- v3) b. 



Similarly the following equations can be deduced 

 (68)... £ {i n + Ir. + }= 6 (2- v3) c, 



(69)...^{in + ir 2 + }=|d f 



and 

 70).,.^{i7cn + O+ }=|-e. 



which give the values of b, c, d, and e. 



In the case of the " temperature curve " already ex- 

 amined, see § 11, r 1 .= 9*02: r 2 = 8*62, etc., and we derive 

 the values a = 62'08 ; b = 3*227 ; c = 8*965 ; d = 0*337 ; 

 e = - 0*890. 



The second (small) co-efficients d and e denote merely 

 small oscillations about the main curve defined by the first 

 pair of co-efficients (large) viz., b and c. 



