540 
Proceedings of Royal Society of EdinhurgJi. 
that the solution of the problem attempted is complete and the 
result very interesting. 
The fifth gives the solution of a problem on which the general 
Theory of Series is said to depend, the problem being the trans- 
formation of 
into an unending series. The numerator, it will be observed, is of 
the order oo : the denominator is of the same order : and all the 
rows of the former except one occur in the latter. Indeed, if the 
first row of the numerator \vere deleted, and the row of the 
denominator, there would be nothing to distinguish the one from 
the other. The subject is best illustrated by a special example in 
more modern notation. Recurring to the extensional above referred 
to as the concluding theorem of the second chapter, and taking the 
case where the factors are of the 4 **" and S*'’ orders, we manifestly 
have 
1^1^2^3^4N^1^2^3^4/5l l%^2^3^4l'I%^2^3^4/5l 4 l^i ^2^3/41*1^1^2^3^4^51 “ ^ » 
from which, on dividing by laj&2^3^4l*ki^2^3^4'^5l 3 obtain 
I 0^1 ^3^4/5 I _ I V 3^4 I J ^lV 3 g 4/5 I ^ I ^iVsAL q 
1 af^c^e^d^ 1 1 \ 1 | 1 1 
Similarly 
I ^^¥3/4 1 _ 1 1 . 1 ^^1^3/4 1 ^ 1 ^1^2/3 1 _ Q 
1 aft^e^ c^ I I ] aft^e^c^ | | af^ ^3 1 
[ ^3 1 _ 1 ^1^2 i . l *^1^2/3 I ^ — 
! ^1^2 ^3 I 1 1 1 ^ 1 ^ 2^3 I 1 ^1 ^2 I 
and 
I ^1/2 1 _ ^ 1. 1 ^1/2! + A = o 
1 e-yn2 1 ^2 1 ^ 1*^2 1 
the last fraction of each identity, be it observed, being the same as 
the first of the next with its sign changed. From the four by 
addition we have 
