SIE J. F. W. HEESCHEL ON A THEOEEM OF DE. BEINEXEY. 
321 
(as V) only occm-, applied all to one and the same power, 0"'; and it only remains 
to develope the function of V we have arrived at in powers of A, bearing in mind that 
v=l+f-f + &c., or l-V=A|-i+|-^V &c.|- 
Now as regards the term > — , it is obvious that it will take when developed the 
V 
form A'*'+”. {a+j3A+yA^+ &c.}, and therefore, that when differentiated n—\ times 
successively with respect to A, it will have the form A'*'''^{a'+i3'A+y'A^+ &c.}, so 
that, when applied as an operative symbol to O'*", the result will be =0. This term 
then may be simply struck out, and we have only to consider the development in A of 
( 
rfv/ V 
Now this is equal to 
( d f 1 / , ^ , {x + n)[x + n—\) 
-\n) 
1.2 
V- 
,) -^- 0 + 0+0 
1 [* + ^] V7ra 
■ _[^n]-| 
- r’ 
1 
0* 
I 
all the terms of which vanish except the first, x being > — 1, and the whole reduces 
— ^ 3 . \ Tt ”” 1 ”1 • 
or to ( — !)”• And we therefore obtain, finally, 
* [x] V 
. ^ nog(i+A) r 
1.2 x'\ A j ^ ’ 
which is identical with that given in my paper above mentioned, equation (6), or in the 
“ Examples,” p. 82. 
J. F. W. Hbeschel. 
Collingwood, April 20, 1860. 
Note by A. Cayley, M.A., F.R.S. 
1 
The above formula (B.), substituting therein for A^ the value V 0*, becomes 
or, as this may be written, 
V“”0*: 
\_x + n\ 
1 
-&c.|0'*'; 
[”— IJl [1] [^— '' ~^[2][a?— 2](n + 2) 
or, inserting a first term which vanishes except in the case x=0, and which is required 
in order that the formula may hold good for this particular value, 
^ ^ ^ V^-&c.|0^ 
Qj_ f r7o 
[J2— 1] )^[0] [a7]n ■ [1] [« — 1 ](m+1) 
V- 
[2] [x—^\{n + 2) 
where the series on the right-hand side need only be continued up to the term contain- 
ing V'"0'*, since the subsequent terms vanish. 
