XLIV XLVI] Inti'n,l,,,-Hon CC-XN ii 



Now the correlation of X and Y corrected for geometrical decadence based on 

 the nth differences is 



In our case the values of n are 5 and 6, and accordingly for the correlation 

 of the death-rates in the first and second years of lift- fit cd from the secular 

 trend, we find 



Males Females Mean 

 n = 5: -'556, -'519, --537, 



n = 6: -'536, -'513, -'525. 



The values found by actually correlating mi and w a after removal of the secular 

 trend by smoothing were* 



Males Females Mean 



r mi m a = - "458, r mimt = - -490, - '474. 



It would appear therefore that Table XLIV will lead to results of the same 

 order as those given by the far more elaborate process of smoothing if we assume, 

 not necessarily that the correlations are geometrically decadent, but that systems 

 of geometrically decadent correlations based on the excess of the three ratios 

 n Ri, n R* and n R 3 , above 4 2/(n + 1) will give an equivalent corrective factor. 

 Both processes confirm each other in indicating that a heavy mortality rate in the 

 first year of life corresponds to a low rate in the second year. We cannot at 

 present measure the relative accuracy of the two methods. 



TABLES XLV XLVI. 

 ft 



Consideration of the Integral I cos n+1 8dd. (J. Wishart, Biometrika, Vol. XVH. 



pp. 6878, 469472.) 



For the complete cos ^-integral, i.e. when the limit = \TT, the value will be found 

 in Table XXVI for n = - 1 to 103. 



For n > 100 other methods must be adopted both for the complete and incom- 

 plete cos ^-integral. Actually if we write a; = sin 2 0, 



f cos MH1 0d0 = ! X (l-x)^x-^dx = ^B x (^n+l,^, 



Jo Jo 



or the cos ^-integral is a special case of the incomplete B-function. Tables of the 

 latter function will shortly be published, but since they do not range beyond 

 n 100, they will not be of service for high values of n. 

 The form of Pearson's Type II a curve is 



and corresponds to symmetrical curves /8i = and j3* > 1'8 and < 3O. 



* Biometrika, Vol. xiv. p. 305. 

 B. II. 



