X 
28 
Mr. Herschel on the 
sion widely differing from his in point of form (though of 
course affording the same numerical results) and which in 
the most important case, where n — — 1, takes a form of 
greater simplicity than any I am yet aware of. 
I purpose then to consider the second member of ( 6 ) as 
developed in a series of powers of ArD (which for the sake 
of brevity we will denote by t). If then we suppose 
/(|= A ,,+iW + A /+&c. 
we shall have 
A - rf-VO 0 . 
* 1.2 X. dt X ’ 
where t = o after the differentiations. 
Now, it is easy to see that 
d x /(n 
dt x 
will, by performing the 
operations indicated, assume the form 
K ftl /• D/(,‘ ) + Dj( e l )+ &c. 
K a being a certain numerical coefficient, depending on an 
equation of differences 
K * + ,,, +I = 0 '+ 1 )- K I>y+I + K 
whose complete integral is 
x,y 
K =C./ C T . 
x,y y ^ y—i x 
— |— • • • • . ( 1 ) 
,x 
1 . 2 , 
7 ^* C x 
Cy— 0 1 
C y being an arbitrary function of y, to determine which we 
have only to consider that x is always, necessarily, unity ; 
and consequently 
(_!)'+' C,.. VL — ,= I 
X I 1 V ' 1.2 ... . {x— i) 
Now, we know that 
(*— l)* 
X 
X — X .. 
■f- &c. 
1.2 .... x 
