Ixxx Tables for Statisticians and Biometricians [LIII 



need of such a table is very obvious, and arises in too great a variety of circum- 

 stances to be specified. 



Illustration. It is required to plot the curve * : 



a; = 14-9917 tan ^, 



y = 235-323 cos^^-e"^^ de-''"^ 6. 



Here log y = log 235-323 + 32-8023 log cos d - 4-5696 log e x 6. 



To cover the whole range of observations we must proceed from 6 = — 45° to 

 ^ = + 45° roughly. It will be found sufficient to take 6 by steps of 3° and 

 ultimately perhaps of 4°. Hence 14-9917 is put on the arithmometer and multi- 

 plied in succession by the natural tangents of 3°, 6°, 9° . . etc. Plus and minus 

 signs are given to these values of x. The corresponding values of y are found 

 in three columns. The first is obtained by putting 32-8023 on the arithmometer 

 and multiplying by the logarithmic cosines of 3°, 6°, 9°, etc. The second is 

 obtained by multiplying (taking the third fiictor from Table LIII) 



4-5696 X log e X -017,4533 = -216,7955 



on the machine and multiplying the result in succession by 3, 6, 9, etc. 



The first column is added to log 235323 = 2-371,6644 and the second column 

 first subtracted aud then added to the result to obtain the value of logy for 

 positive and negative abscissae respectively. The antilogarithms give the ordinates. 



Another problem sometimes arises given x to find y. For example : In the 

 above curve the mode is at 117-999S cms. of stature and the origin at 113-8228, 

 thus the distance between them =4-1770 cms. or since the working unit of 

 x — 2 cms. and the positive direction of x is towards dwarfs, the mode is at 

 X = — 2-0885. Required to find the maximum ordinate ym„. We have 



tan e = - 2-0885/14-9917 = - -139,3104, 

 whence by a table of natural tangents 



e = -T 55' -851265, 

 = -r 55' 51". 

 The log cosine of this value of 6 is 



1-995,8962. 

 Table LIII gives us : 



7°= -122,1730 in arc 



55'= -015,9989 „ „ 



-851,265' = -851,265 X -000,2909= -000,2476 „ „ 

 Hence ^=--138,4195 „ „ 



* See Phil. Trans. Vol. 186, A, p. 387. Pearson's Type IV frequency curve fitted to the stature 

 of 219'2 St Louis School Girls aped 8. 



