20 



KARL PEARSON 



Table II. Values of the co-functions. 



I owe this preliminary table of ui-functions to the kindness of Dr Alice Lee. Much more 

 elaborate tables will have to be calculated, if as I anticipate the <o-functions are found valuable for 

 other purposes. The present table suffices to indicate their general numerical character, and enables 

 one to calculate some of the quantities needed in the present memoir. 



graphical construction. The fact that the central ordinate of the sixth curve is 

 almost identical with the ordinate of the fourth curve at r = I, seems conclusive as to 

 the general accuracy of the process. 



The above test of the general accuracy of Mr Blakeman's graphical work is only a 

 part of the still more sufficient test that in the seventh curve the graph and the 

 co-expansion practically coincide. See Diagram VI. After r = 5l the two curves 

 cannot be distinguished, and between r = and 31 the deviation is probably as much 

 due to the neglect of higher w-functions as to errors in the graphical treatment. 



Another method adopted by Mr Blakeman for testing the accuracy of his 

 graphical work, especially at the end of the range, was to obtain expansions to 

 (f>n (^)> when r does not differ much from nl, = nl — £, say, where f is supposed small. 

 If f n (£) = g/>„ ((nl — f ) 2 ), then generally for £ small : 



N /£\(»-3)/2 



.(xxxix), 



/.(*) = 



\*+*„»-iJ n \l 



IJJ,...I n 



where 



? (J 2)" 



= T cos«0d0=r(£(w+i))r(f)/r(£(n+2)). 



