156 
then by Hermites Law, each of the n? ¢ (nu | ©,, ©.) 
4 u 
—— 
n n 
can be expressed as a linear homogeneous function of xi. 
We must now divide our discussion into two cases (a) 
n=1 (mod. 2) 
ort. fe | 1, 9) 
X a (u) =f, (1,2) II «eq u 
n p=0 BP a 
ie { n2 =| { ee U] — j| 
=f, (0,, o,) Ppa Las ae | 5) UP ‘be (a0, + == 2) | 
ase it ao, — HO, 
IT o(u——Fa— | 41, &2), 
as) 
Whence X , (u) is a o-product of n? factors whose residual sum 
n2 
; ea 
— aw, + — Oz 
And hence can be expressed as a linear function of x; defined in equation (4). 
* There are n? quantities X g (u). 
n2 
We have now shown that we can express the n? quantities X , (u) as linear 
n2 
homogeneous functions of xi, and also the n? quantities o (nu | ©,, ©,) as linear 
A pb 
homogeneons functions of xj, whence we can express X _ (u). as a linear homo- 
n? 
a ED 
(b) n = 0 (mod. 2). 
In this case, 
nei ( 
X ,, (a) = (0, 4) iW ‘a thee) , (Equation (1) 
a Ll pws 
n2 -E=0 no >? Te 
2 f a 1 a her { a rm: ) Yn as 
files ne] SU Wea ek ee me th Oe 2 i ee 
2 
*Klein: Vorlesungen, etc., Vol. II, p. 264, 
