\\n 



Putting a? = 0, 1, '1, \w lind \'r the bournlin^ ordin.-ttcs of tin- .n 

 (274n._ 2 - :._! + 274;,. - 120//. fl 



1 



I -JO/, 



1 

 l'2()l, 



1 



*- 1 " 86n " 



(9). 



-T.V/M ( 3fWs 2 ' 



120/t 



In a precisely similar manner and to the like degree of approximation (i.e. 

 vanishing of fifth differences) the differential coefficients of the y's or again their 

 differences can be found in terms of the sub-frequencies. 



Formulae (2) or (6) may also be occasionally of service for readjusting the 

 frequency on subranges; for example, to half-subranges, etc. More ample formulae 

 are given in an Appendix to this Introduction. 



(2) On Methods of Interpolation into Tables of Double Argument. 



We will use z for the tabled function, h and k for the arguments. (= 1 <f>} 

 will represent the proportion of the unit of the argument h, and ^ (= 1 -^r) will 

 represent the proportion of the unit of the argument k, and z 6fX the value of the 

 function z, at the point of the table defined by 0, %. The units of the arguments 

 of h and k may be anything whatever, they are only represented as equal in the 

 following diagrams for convenience. Of course d and ^ must be computed having 

 regard to the absolute size of the units. 



Fig. 1 represents the general case, when we need to interpolate into the body 

 of a table. 



Here, if the central differences are not tabled, we find them at once from the 

 formulae, where r refers to the h, s to the k axis: 



S 2 z r<li = ,.+,, + z r -i >s - 2z r>li "I 



where S 2 refers to differences with regard to h, and S' 2 with regard to k. 



For most practical statistical purposes, it is adequate to use central difference 

 formulae which proceed only to 8 2 and 8' 2 , i.e. formulae correct up to and including 

 third differences*. Very often it is not needful to proceed even to S 2 and S' a , and 



* For fuller treatment see Tracts for Computers, No. III. Cambridge University Press. 



B. II. 



