1 
414 The liicomjjlete Moments of a Normal Solid 
1 1 - 
Similarly by putting = g we have p = ~ and 
Vl-J9 ? 
r„ l-f r„,_, /I - r/Y _ 1 T,,{qx) 
«! 5" 2(n-2)! V ; 2^2!(n-4)! 5" ??,! 
The following developments may be easil}" proved by expansion and simple 
addition : 
T^{x) x"^T,,_,{x) x'T,^,{x) ^ x^ ' Tn{l) (1) x^-^ 
n\ 2(/i-2)! 2^2!(n-4)! n\ 2(«-2)! 2^2!(?z-4)! 
(74), 
Tn{x + a )^x^^ x^^-'T,(a) x ^^-'T,(a) 
n\ n\ (n-1)! 2!(m-2)! ^ 
Since we have the binomial coefficients multiplied into terms of the type 
(a) we may write this symbolically 
T,Xx + a)=[x + T{a)Y, 
with the convention that the x'a will have indices and the jT's suffixes. Clearly we 
may also write 
T,,{x^a) = {T{x) + a)\ 
Similarly 
{X + aj' Tn {X) Tn-, {X) . ( 
n\ n] (n-l)!l! (/i-2)!2! ^ ^' 
Therefore (x + ={T{x) + m (a)}" 
= \m{x) + T{a)Y\ 
with the same convention as before as to indices and suffixes. 
2' (x) 
The following is of interest. Write 6n = " . , then 
° 111 
d6n_ d a;""' 
~d^~dxW.~ 2.(h, -2)! ^ 2^2!(/i -4)! 
t</ cty «X/ 
"'(h-1)!~2.(?j-3)!''"22.2!(ji-5)!'~"" 
=J;-4)r«- 
Since rn = i:T„-,-(H-l)T„.„ 
1 / dOn d^d„ 
= - { X 
n 
/ d£n _ rt^A 
V dx dx" ) 
11 \ 
If 
= - [ X — 
n \ 
= ^-(x-^\ (a>-^\^^ 
11 \ dx) \ dx) n — 1 
d\ d 
dx) dx 
A 
dx 
1 / \» 
n\\ dx. 
.1 as). 
