76 
ME. W. H. L. EUSSELL ON THE CALCHLUS OE SYMBOLS. 
Section II. On some General Theorems, 
I shall now give some theorems in general differentiation and expansion. 
Since 
to n factors, we have 
(?•’")'=(?•")(?")(?•")■•• 
(p-'f) 2 )(t— 2ffl+2), 
(p-T)".I = ^.(T-l)(.r-3)...(.r-2»+l); 
whence we easily see that 
.r(x-l)(.r-2)(T-3)...(ff-2«+l)=g“«(y.x)V”-(p.r)", 
^(t-1)(t-2)(t- 3).. , (x-2»+l)=?(ix) >- (ix)’, 
whence we shall have 
X 
dx^^’’ \x dx 
1 d\^ d\^ 
\n , 
X dx 
IL 
If we equate the coefficients of in {l-\-zf”={l-\-z)”(z-\-iy, we have 
2n{2n— l){2n—2) .,.{2n—r+ 1) n{n — l) [n — 2)...{n-^r + \) 
1.2.3...?- ~ 1.2.3. ..r 
, — 2) ...(/i — r + 2) , n{n — 'l)...{n—r + S) n{n—\) , 
^ i 9 A 1 •W'-j- 1,2,3, r — 2 ' 1 2 < 
1.2. 3. ..?•-! 
27r(2‘r — 1)(2‘T — 2) .... (2t — r ff”!) 
='r(‘;r — 1) . . . {'7t~r-\-\)-\-r'z{‘7r — 1)(t — 2) . . . (7r“f-j-2)T 
?— 1 
ff-r— ^‘y(7r~l)(T--2). ..(t— f+3). 7r('r~l). 
Hence, smce 
we have 
?— 1 
+r'—^f 
whence we find 
1 \’—2 /I \2 
\d.x2/ 
* It has been pointed out to me that this theorem might be more shortly proved by applying Vandeb- 
moiiDe’s theorem to the equation (2D)’■=(D + D)^ I have retained the demonstration in the text merely 
r the sake of the method. 
