234 
Oti Con-ections for Monient-Coefficients 
Since dZjdo; — — y,\t follows that 
y/„ = _ ^3 = ( i37„; _ 1(53;,/ + 137,,; _ 53,,; + 19,,;) 
Nhi N 
■ijp = -j ' = QQi^ [lS7n'j, - 16o«'j,_i + 137/i'^,_o - uo». -r ^,-41. 
These results enable us to determine approximately the terminal ordinates of 
the frequency distributions given by sub-frequencies, and to discover how nearly the 
frequency curve comes to zero at the terminals of the range. Similarly the small- 
ness of the quantities «,,) fa, t'4> ^^'5 ii^nd ^*2, ^3, ^4, ^5 marks the character of the 
terminal contact. At the same time the reader must remember two points (i) that 
the terminal frequencies if small may be subject to large probable errors and 
(ii) that we have supposed i/ = 0, when x = and x = Xp, the terminals of an integer 
luimber of subranges. It is extremely unlikely that the frequency curve would cut 
the variate axis exactly at such places. Hence on both counts, (i) and (ii), we must 
not anticipate in actual practice that «i and will vanish at x = x^^ and x=- Xp for 
non-abruptly terminating frequency, unless we know a, iiriori the terminals of the 
range and have chosen our subranges to fit this knowledge. 
(3) The next stage in our work must be to table the values of 
dZ'jdx, d'Z'/dx\ ... d'Z'jdx^ 
where Z' = Zx^ at the two terminals of the range. We may do this for s=0, 1, 2, 3, 4, 5. 
The theorem of Leibnitz provides the needful expansions which are 
dZ' _ JZ ^ . 
— Ob 1 ~\~ sec Ju ^ 
dx ds 
d^Z' d^Z ^ d'Z . , „dZ . ,^ 
dxi-^ dx' dx^ dx ^ ' ^ ' 
- = -TV + osx'-^ -r— + 10s (s - 1 ) x'~'- ^ + 10s ( s - 1 ) (s - 2) x"-^ -r-^ 
dx^ dx" dx* dx- dx- 
dZ 
+ 5s (s - 1 ) (s - 2 ) (s - 3) X'-' -^^ + s (s - 1 ) (s - 2) (s - 3) (s - 4) x^-"Z, 
d'Z' , ^ .,d'Z „ . ... . .d'Z 
= 21s (s - l)x'-' + 35s(s- 1) (s-2)^'«-^ + 35s(s-l) (s-2) (s - 3)^, 
+ 21s (s - 1 ) (s - 2) (s - 3) (s - 4) af-' ^ 
d Z 
+ 7s (s - 1 ) (s - 2) (s - 3) (s - 4) (s - 5) -j- 
+ s (s - 1 ) (s - 2) (s - 3) (s - 4) (s - 5) (s - 6) Z, 
= 126s (s - 1) (s - 2) (s - 3) x'^-* ^+ 126s (s - 1) (s - 2) (s - 3) (s - 4) ^"^^ 
+ 84s (s - 1) (s - 2) (s - 3) (s - 4) (s - 5) ^ + etc (vi). 
d. 
