382 
Miscellanea 
But «/o = o, and if there be high contact we can form all the ordinates «/o , j/i , 3/2 y -i-, y -21 ••• 
by using the expression given for a, hence, if X be the distance of from the vertical about 
which moments are calculated we have as the general term for the «th moment 
[2 1 34A'» - 1 1 6 {( A' - 1 )" + ( A' + 1 )»} + 9 K A' - 2)» + (Z + 2)"}], 
which after reduction becomes 
"1920 A- . 80 . ^^„-2+^^(»-l)(^^-2)(^^-3) _ _ 
2 ! 4 ! 
and this will be found to give Sheppai-d's adjustments. 
The values of 6, c, d and e are given above as they may be useful in some cases other 
than those with which we are now dealing, e.g. for some work when jjarabolas are being 
used. They enable us to find, for instance, 
3/_2 = Y^{16894_2 + 684^_i-746Ao + 364^1i-7U2}, 
which, by the way, if applied to all terms of the series also give Sheppard's adjustments when 
high contact is assumed. 
The method in a slightly modified form might be applied to enable us to deal with the cases 
in which the curve rises sharply from zero, but the difficulty is that in all such cases the actual 
starting point makes a very great difi'erence to the result we obtain. In fact it is possible 
if we assume the curve to start at the beginning of one of the groups to obtain a negative 
ordinate at the middle point of the group and the result is in consequence extremely unsatis- 
factory, though it gives some idea of where the curve really starts. 
rr + i 
Using the same notation as above, viz. | 7/dx = A,., and also the same method, the 
following formulae can be obtained : 
If y3: = a + 6.^'-|-c.«^^ + c?.r^-|-e.r'', 
i/o = j^{21344o-116(4-i + ^i) + 9(4_2 + ^2)} as above, 
9/^ = :^~{-7lA2 + 20UAi-26A,y-26A^, + 9A2'f XII., 
y.2=^~{16mA, + 684-UQA, + 36iA^,~71A_^} XIII. 
If y = a + bx + cx^, 
yo = ^^{-A_, + 2GA,-A,} XIV., 
^i = ^{23^i + 2^-^_i} XV. 
In order to apply these formulae to the problem of correcting moments we first find the 
ordinates corresponding to the given areas by means of Formulae XI. to XV. and then use 
Formula X. or some similar simpler ex^jression such as 
j *.yx«?.*--=~ {26^0 + 213/1 + 25^/2 + 24^3 + 242/4 + ...} XVI., 
