190 ON SKEW SURFACES, OTHERWISE SCROLLS. [339 
where II denotes the product of the terms belonging to all the triads of the m roots, 
the result will be symmetrical in regard to all the roots ; and replacing the symmetrical 
functions of the roots by their values in terms of the coefficients, we have the required 
relation between (ft t), ft &>). 
II contains £ [m] 3 terms, whereof |[m — l] 2 contain the ra-thic functions (A 1} B 1} C x , A) 
of the root 0! ; that is, the form of II is 
(A, fx, v, p)è [w]! (0 1} 1)W(U, l)* [m]3 .. ; 
or, when the symmetrical functions are expressed in terms of the coefficients, the form is 
(A, fx, v, p)èM 3 (f, 7], ft, <w)£ [m]3 . 
Now the above-mentioned determinant is divisible by (0 3 — 0») (0i — 0 3 ) (0*— 0 3 ), or IT is 
divisible by II - ft) - 0 3 ) (0., - ; and since this product contains (3x|[m] 3 =)|[m] 3 
linear factors, and the product Ç(0 X , 0 2 ,...0 m ) of the squared differences of the roots 
contains (2x| [m] 2 =) [m] 2 linear factors, so that we have 
where 
and consequently 
n (0 1 - 0 a ) (0\ - 0 3 ) (0 2 - 0 3 ) = {£ (0„ 0 2 , .. 0 m )}i^», 
Ç(0 1} 0,, .. 0 m ) = Disct. = (ft 77, ft w) 2(m_1) , 
n(ft-00(ft-*.)(*.-*.)-(£ v, ft «) [m " 1]2 , 
so that, omitting this factor, the remaining factor of II is of the form 
(A, fx, v, p)i [m]3 (|, y, ft ; 
but the determinant vanishes if 
A, fx, v, p = (A u B 1} C 1} A), (A%, A, G,, A), (A 3 , B 3 , C 3 , A), 
(A, fx, v, p) = (A, B, G, D), 0 = 0 lt 0,, or 0 3 ; 
it follows that the product II contains the factor 
(A£ + pci] + + pco) ; 
or omitting this factor, and observing that 
i [w] 3 - [m - l] 2 - £ [m] 3 = £ [m] 3 - [m - l] 2 = £ [m - l] 3 , 
the remaining factor is of the form 
(ft v, ft «) itw_1]3 ; 
or we have finally 
$ (m 3 ) = i [m — l] 3 , 
which is the required expression.