210 
ME. A. CAYLEY ON THE DOHBLE TANGENTS OF A PLANE CTEYE. 
and I consider the expression 
the general term of which is 
— 2S4*1 — s)} 
X r [n-lJ[»z-2]“[^-27 ^ 
L + &c. J 
or as this may be written, putting q=n—'6—l^ 
^"l“2d-5, (f — 2^+1 — s)} 
r [^„2]t^-27 1 
I 4- &c. J 
30, I assume 2, we have then and therefore a+/3<g. 
The general term of the series in { } is 
i-f 
a[^- 
• 2 — — 2 — 
where the terms for which w— 2 — -S- is negative are to be excluded, or what is the same 
thing, the series is not to be continued beyond S'=?z— 2. But obsening that [ 5 ']^ 
vanishes for S^>§', that is, S'>w— S— 1 , it is in fact the same thing whether the series is 
continued indefinitely or only to the term for which ^■=^^ — 1 , and we may consist- 
ently with the condition 2 , continue the series as far as 1 , except in 
the case ^= 0 , when by doing so we include the term corresponding to — 1 , which 
in virtue of the condition ought to be excluded. The expression for the term in question 
is (—)”■'[— ; hence if the sum of the series continued to the proper point is S, 
the sum continued indefinitely (in the particular case S= 0 ) is S+(— 
but in every other case the sum continued indefinitely is simply S. And by a well-known 
theorem in finite differences, the sum continued indefinitely is in fact zero. That is. 
except in the case B= 0 , we have 
S=0, 
but in the excepted case 
or observing that a+/3(=r— S) is in this case =?% and transforming the factorials, we 
have 
s=(-r-’W“C57, 
or substituting for a and (3 their values, 
S = ( — )”"’'[5]*[r — S]’""*. 
