256 
ME. W. H. L, EIJSSELli ON THE CALCULUS OE SYMBOLS. 
Take for a divisor, and proceed as follows: — 
— TT- ) +^(T"+7r + 1) — 
g^+e 
-j- 
It is easily seen that the remainder may be written 5r(‘T— l)(g'— x). 
Hence, taking f — w as divisor, 
g— ’r)g"+f— 
g(5r+l)— w" 
g>(T+l)— ■r’* 
Hence ^ — ir is the highest common divisor of — ‘jr® and 
Again, to find the highest common internal divisor of f^^(^+l)+f('^^+?r^)+7r^(x — 1) 
and f'T — 2^Vd-g>(7r^-j-5r)-f-7r'*. 
The first of these quantities is equivalent to (tt — §{'7r^ 
We take pV+^(5r^+‘r)+^r'^ for divisor, and proceed as follows: — 
g>Vd-|'(‘T^-j-^r)-l-T^)|>V— 2^V-|-^(T’^d-7r)+T'‘(^ — (■r+l) 
H" Tt) + 
— f^(^^+3'r)d-g>'r4-5r^ 
— g’^(^’*d-3‘r) — g>(x®+3‘^^+2'5r) — ‘r’^(Td-l) 
f 4" 3 ^^ + Stt) -j- 
The remainder is equivalent to (7p4-r+l)(g®'+’*'^)* 
Hence, taking for a new divisor, 
gr+^r" )gV+^(9r-+^r) d-Tr"* (^+ 1 
gV+g?r" 
gT+T^ 
gT+T^ 
Hence gT-h-r^is the highest common internal divisor of 
^{tc^ -\-'7c^)-\-'7r\'7r — 1) 
and 
gV — 2gV-}-g(’r’*4-^)-l-^^ 
It is evident, as was mentioned in the introduction to this memom, that this process 
is equivalent to finding the conditions that two linear differential equations may have a 
common solution. 
I shall next proceed to find the general term of the continued product 
( § + f + S2^)(g + ^3’^') •••(? + 
