HlMAJiAYAS ON THE PLUMB-LINE IN INDIA. 
63 
1 
40 
9«, 
in 
cases. 
1 
8 
in 
3 
cases. 
1 
7“’ 
in 
2 
cases. 
1 
1 
6 
in 
case. 
1 
1 
in 
case. 
1 
1 
in 
case. 
1 
1 
in 
case. 
49 
Let us take, then, the extreme case of the first 40, viz. a=38° and (p=-ot. Hence 
j-fJ(j&=40° 6', and a+(p = 42° 13' ; and 
cos®^ 0 
Greatest value of — ^ — =2*746 
sinis 
Middle . 
Least 
=2-574 
= 2-418 
Difference. 
0-172 
0-156 
Ratio to 
middle value. 
T^-th. 
l^th. 
Hence in the most unfavourable of the first 40 cases, viz. that in which <p is exactly 
=^Q5, the extreme values will depart onlyi\th and^^g'^^i from the middle value: 
and the successive pairs of values on the two sides of the middle will differ less and 
less as they approach the middle. Suppose that there are n such pairs, and that they 
form two series in arithmetic progression. 
— ^ 0 and 0 +::^* 
15 ' 16 
The sums of these two series are — and -l--^.^^-^5 
and hence the average error or the mean of these 
1 j-n — L L 
“ ^15 16/ 2 - 960 2n+\~ 960’ 
when n is indefinitely increased. 
Hence the probable error in this extreme 40th case is in defect, and is about 
— (/o oth part of the whole. The probable error in the 9 cases beyond the 40th will 
be greater than this ; but for the 39 cases before the 40th far less and less. 
K 2 
