380 
Proceedings of Royal Society of Edinburgh. [sess. 
its order-number cannot exceed n(n + 1) - n i.e. n 2 , Joachimsthal 
concludes that Y 1 and U can only differ by a factor dependent on 
the a’s. He thus has the two results 
J • A\A\ . . . A 2 = A(u y , u 2 , . . . , u n+1 ) • Y 1 
and V, = { • U = ( • A X A 2 . . . A n [l , 2 , . . . , n+ 1] 
where £ is a rational function of the a’s : and by combining the 
two there is deduced 
J 
MU 
n _j_ i j . , z^ 2 , . . . , u n+1 ) ' 
-A]A 2 * * * ^n 
At this stage, we are told, the investigation rested for five years 
until the publication, in 1855, of Borchardt’s paper in the Berlin 
Monatsbericht. Taking a hint from this, Joachimsthal, in order to 
determine £, multiplied both sides of this result by the product of 
all the denominators occurring in the diagonal of J, and then put 
u l = - a Y , u 2 = - a 2 , . . . , u n = - a n . The left-hand side was 
thus changed into 
1 0 .... 0 0 
0 1 .... 0 0 
0 0 .... 1 0 
I 1 I ! 
(«, + «a+,) 2 (« 2 + u n+1 f ' ‘ (a„ + u n+1 f 
or 1 ; the second factor of the right-hand side, being equal to 
( fa j — U n + 1)(^2 — ^n+l) * * * fan ~ ^n+ 1) • ^fa\ ) ^2 > * * * > ^ n )) 
was changed into 
l) n (a 1 + u n+1 )(a 2 + Un+ 1 ) . . . fa n + u n+1 ) . (-iy- n(n ~ 1) A(a 1 ,a 2 , . . . , a n ) 
and the third factor [1,2, . . . , ?z+ lJ/AjAg . . . A n into a 
fraction with the numerator 1 and with the denominator 
(a 1 - a 2 )(a 1 - a 3 ) . . . (a 1 -a n ) fa l + u n+l ) 
. (a 2 - a 1 )(a 2 - a 3 ) . . . (a 2 - a n ) (a 2 + u n+1 ) 
. (a 3 - a 1 )(a 3 - a 2 ) . . . (a 3 -a n ) fa 3 + u n+1 ) 
. (a n - a 1 )(a n - a 2 ) . . . (a n -a n _ 1 )fa n + u n+l ) 
or 
< - l) in(n_1) A(a 1 , a 2 , . . . , a n ) 2 • (a l + \ + i)(fl 2 + %) . . . fa n + u n+1 ). 
