172 
Proceedings of the Royal Society of Edinburgh. [Sess. 
A glance down the columns of Table VI will show that the successive 
differences of the values of change very slowly, so that over the interval 
between any two we may assume a linear relation between and r. 
Suppose, for example, that rj A = br — a. Then h and a are to be determined 
from the equations 
br x - a = rji 2 ) 
hr 0 — a = p 2 J ’ 
where r 0 is the value of r corresponding to p and r Y the value of r 
corresponding to rj v the value of *7 immediately above p. 
Thus 
6 = ^1 
2 -fr 
and 
a+p 1 
The integral then becomes 
This is the area of the last element which contains the infinite branch. 
In order to be able to apply Weddle’s Rule for integration systematic- 
ally throughout, it is necessary to group the numbers in sets of seven, the 
seventh of any one set being the first of the succeeding set. The very last 
number which is the p 2 in the formula lies outside the last set of seven. 
Hence the positions of the numbers which are to be chosen as the different 
values of p 2 are to be represented numerically by any number of the form 
Qn-\- 2 where n is an integer. There will then be n groups of seven, for 
each of which the integral is to be summed. This summation, when 
properly worked out, will measure the angle between the radius corre- 
sponding to the first number and that corresponding to the last number 
of the set. Thus n points are obtained lying on the portion of the path 
or ray between the vertex and the surface, that is, on half the ray. The 
(w + l)th point will be given by taking into account the last element 
involving the infinite branch. 
The method will be made clear from the details of one of the cases. 
Let n = 9 ; then there will be nine points given by the Weddle summations, 
and the value of vj 1 chosen as the p 2 will occupy the"( 6 'ft + 2 )th position in 
the tabulated figures, that is, the 56th place. For this position r— m 725, 
