378 Proceedings of Royal Society of Edinburgh. [sess. 
being dependent on the solution of the sixth-degree equation 
at? In? eg 2 = , 
(a + u) 2 ( b + u) 2 (c + u ) 2 
the relation between four of the six roots is evidently 
1 
1 
1 
(“ + “ l) 2 
(c + u x ) 2 
1 
i 
1 
(a + u 2 ) 2 
(6 + V 2 
(c + u 2 ) 2 
1 
1 
1 
(«+%) 2 
(6 + w 3 ) 2 
(c + « 8 ) 2 
i 
1 
1 
(a + u^ 2 
(i + » 4 ) 2 
( c + « 4 ) 2 
Joachimsthal knowing this, and having obtained by an entirely 
different process the result 
(a + w 1 )(& + u 2 )(c + u s ) ^{a + uf{b + u 2 )(c + u 4 ) 
+2 
(a + u-^(b + u^)(c + u 4 
+ y 
(a + u 2 ){b + u z )(c + m 4 ) 
= 0 
where the sign of summation refers to permutation of the u’s, is 
naturally led to inquire into the connection between the two 
results, and to extend the inquiry to the higher cases of the same 
kind, including, therefore, the evaluation of the determinant 
1 1 1 
{a 1 + zq) 2 
(®2 + “l f 
(a n + u l )‘ 
1 
1 
1 
(«1 + m 2 ) 2 
(« 2 + 
(«» + u 2 y 
1 
1 
1 
K + U n+1 ) 2 
(a 2 + u n+1 ) 2 ' 
(a n + u n+lj 
The investigation of the determinant occupies the fifth and sixth 
sections (pp. 164-169) of his paper, the third and fourth being 
devoted to obtaining the other form of the resultant 
2 
+ 
1 
+ 2 
2 
1 
(«i + u 1 )(a 2 + u 2 ) . . . (a n + u n ) + u 1 ) . .. (a n _ x + u n _ x )(a n + u n+1 ) 
1 
(a 1 + u 2 )( a 2 + u 3 ) . . . (a n + u n+1 ) ’ 
