400 Proceedings of the Royal Society of Edinburgh. [Sess. 
one is the derivate of the other ; but, unlike Cayley, he makes the said 
functions of different degrees, namely, 
V” + + C 20^ n “ 2 + • • • • + C «0 
C Q1 X 6‘ii^ + C 21 X + .... 
Following the ordinary division-process for expressing the ratio of the 
second function to the first as a continued fraction of the form 
a p, 
x +i + & 
X - 
Ps 
1 + . 
and denoting the remainders in order by 
— | c 02 x n 1 
Cm ( 
„n — 2 I „ 3 
n rv^ 2 I /. /yjl 3 I /. /y.1l 4 i 
'13' 
^ f f> /yW* 3 | ^ 4 | ^ qjtfl 5 j 
S ^04^ r ^p4*^ “ ^ 24 ^ i • 
C 01 C 03 * 
he finds that 
P 0 = X 
Coo ^O.r^O, r+1 C n 
■'O , ‘in— 2 
The second suffix of any one of the new c’s is seen to indicate the remainder- 
function to which the c belongs, and the first suffix the position of the c in 
that remainder. To obtain expressions for these in terms of the original 
two sets of c’s it is taken for granted, and with reason, that as a result of 
the process we have generally 
C r ,s C 0 , 
s-l'-'r+l, s-2 — , s- 
! G r + 1 , 5-1 — 
-'O , s-1 °0 , s-2 
C r . 
r+1 , s-1 ^r+ 1 , s-2 
( 1 ) 
By using this twice upon itself, so as to lower the second suffixes of the 
first column, there is found 
'■'0 , s-2 
C 1 , s-2 
'0 , s-3 
J 1 , s-3 
( 2 ) 
'r+2 , s-2 C r + 2 , s-3 C r + 1 , s-2 
where the second suffixes are now s — 2, s — 3. A page is then occupied in 
ridding (2) in the same way of the elements which have s — 2 for a suffix, 
the result being 
C 0 . s— 4 
"1 , s— 4 
'2 , s— 4 
'0 , S-3 
'2 , s-3 
r+3 , s-3 C r+ 3 , s-4 C r+ 2 , s-3 
( 3 ) 
^r + 2 , s— 4 ! • 
