166 Proceedings of the Royal Society of Edinburgh. [Sess. 
at the upper end of the column a number less than seven, then the quadra- 
ture may be effected by the appropriate formula. Although this is less 
accurate than Weddle’s formula, it affects the evaluation of a very small 
part of the whole and the error involved is of no moment. 
The following are the areas of the successive portions for 14: — 
Limits of Values of p. 
Area. 
14-1 to 14-5 
2-914 
14-5 „ 17-5 
7-250 
17-5 „ 20-5 
3021 
20-5 „ 23-5 
1-626 
23-5 „ 26-5 
•816 
26-5 „ 28-5 
115 
141 to 28-5 
15-742 
The area of the part which passes off to infinity has now to be 
estimated, namely, 
f f(p)dp 
JJ(p*-y 2 ) 
between the limits p 1 = rj(l and p 0 = r/, where e 1 is a small quantity. 
Writing p = rj(l -f e) we, have 
dp = rjde 
and 
Hence 
- r) 2 ) = 7)(2e + e 2 y = rij-2e(l 
f(p)=Ap)i- e vf'(p) 1+ • • • 
I'i pAy)t-<(g)jLt *+$£_ . . )de 
Jo J2e \ 4 8 4 / 
l [ e \ e -i - $ e +i + , . ^_^Mx(V-K + ■ • .)de 
J2 Jo J2 Jo 
J(V\ )(2ep - ^ (fe-p - . . .) 
J 2 
J2 
— f( v ) ./o e f\_ i c _i/(P)r7 e i\ 
/Wi ' /Jei \ 1 1261 3 f ( Pl ) 1 
In the particular case chosen above 
12 ' 
r, \ /(Pi) 7 ! 6 ’25 OT 
f(p,) = 19-75, J V v 3 { ‘ = - — - — , e = — . 
f(p) 19-75’ 14 
r = 
19-75 
J 70 
1- 
1 1 
1680 + 3 
25 
1975 
2-369. 
Hence 
