406 Proceedings of the Royal Society of Edinburgh. [Sess. 
where necessary, as to bring about axisyminetry.* Lastly, he tries 
(pp. 43-46) to justify Cayley’s rule for calculating the coefficients placed 
inside any square of the chain. 
Borchardt, C. W. (1859, November). 
[Vergleichung zweier Formen der Eliminations-Resultante. Crelles Journ. T 
lvii. pp. 183-186 ; or Gesammelte Werke, pp. 145-150.] 
The problem here is exactly the same as Hesse’s of the previous year. 
Instead, however, of multiplying S by £(/3 l5 /3 2 ), he preferably multiplies 
, 0 2 , cq , a 2 , a 3 ) by S, thus obtaining 
1 ft ft* ft» ft 4 
% a \ a 2 ’ 
<HPi) &<Mft) ■ 
1 ft /y ft* ft 4 
a 0 a x a 2 a 3 
^(^2) 
1 otj cij 2 a x 3 a-j 4 
5 0 b l b 2 
= 
H a i) a M a i) a iV( tt i) 
1 a 2 a 2 2 a 2 3 a 2 4 
■ \ \ \ ■ 
If/(a 2 ) a 2 xjj{a 2 ) a 2 2 if/{a 2 ) 
1 a 3 a 3 2 a 3 3 a 3 4 
• ■ K h 
H a s) a sH a s) a 3V( a 3 ) 
Now, E being Euler’s product of differences, the first determinant on the 
left is resolvable into 
* £ 2 ( a 15 a 2> a 3) * E > 
as was first observed by Rosenhain in 1845 (Sept.) ; and the determinant 
on the right is resolvable into 
| &( P V P 2 ) • < KPl ) • <£(&) } { C’( a l> a 2> a 3) • «A( a l) • H a 2 ) • ^( a s) } • 
We thus have 
ES = | <£(&) • <A(&) | { *A^ a i) • H a 2) • H a s) | > 
= «3 2 E . 6 2 3 E , 
and .\ 
S = a 3 2 b 2 3 'E , as before. 
* The expression for R 4 , 4 , though now given more accurately than before, is still dis- 
figured by at least ten misprints. 
LIST OF AUTHORS 
1844. 
Cayley . 
whose writings are 
PAGE 
. 396 
: herein dealt with. 
1856. Cayley . 
PAGE 
. 402 
1845. 
Heilermann . 
. 398 
1858. Zeipel . 
. 403 
1848. 
Cayley . 
398 
1858. Hesse 
. 403 
1852. 
Heilermann . 
. 399 
1859. Bruno . 
. 405 
1855. 
Bruno 
. 401 
1859. Borchardt 
406- 
{Issued separately May 7, 1910.) 
