274 
On the Systematic Fitting of Curves 
The values of the coefficients 7 are as follows* : 
7i = 
•083,3333 
7/ 
= 
•041,6667 
72 = 
■041,6667 
72' 
= 
•041,6667 
73 = 
•026,3889 
73' 
= 
038,7153 
74 
018,7500 
74' 
•035,7639 
76 = 
014,2692 
7s' 
•033,1918 
76 = 
011,3674 
7/ 
= 
•030,9989 
77 = 
•009,356o 
/ 
77 
•029,12o3 
78 = 
•007,8925 
78' 
•027,5110 
79 = 
•006,7858 
7/ 
•026,1066 
7io = 
•005,9241 
7io' 
•024,8732 
7ii = 
•005,2367 
7n' 
•025,7807 
7l2 = 
•004,6775 
712' 
•022,8052 
7l3 = 
•004,2150 
7l4 = 
•003,8269 
Now the Euler-Maclaurin formulae possess marked merits and defects : 
(a) The correction terms being usually small, they equally weight all the 
observations in the bulk Ac and At oi the formulaf. This is of much importance 
when the observations are liable to considerable error. 
(h) They will give the best possible results if we go to the complete system 
of differences for the jp ordinates. 
But : 
(c) To do this involves in most cases very great labour. The coefficients 7 
do not converge very rapidly, and the A's in many practical cases, especially of 
frequency, do not get rapidly small. 
(d) If we stop at the third or fourth difference, then the 7 coefficients are 
not the best coefficients by which to multiply the successive differences, but the 
best coefficients differ considerably from these if p be not very large. 
In order to get over (c) a number of formulae have been used which depend 
upon the number of ordinates used being a multiple of 2, 3, 4, 6, etc. Thus we 
have the following rules : 
Simpson's Rule i2p elements). 
zdx = Ih[z^-¥2{Z2 + Zi + ... + Zip-^ + Zip 
+ ^{Z^ + Z3 + ... + z^p^^)] (7). 
* Calculated from the formulae given by De Morgan ; Differential and Integral Calculus, Art. 61, 
p. 262. 
t Except in the case of the first and last ordinates of Ac, which clearly can only be given half 
weight. 
