MR. W. H. L. RUSSELL ON THE CALCULUS OE SYMBOLS. 
77 
I now come to the theorems respecting expansion, which I mentioned in the begin- 
ning of the paper as analogous to the binomial and multinomial theorems in ordinary 
algebra. 
To expand in powers of x, where $(Tr) is a function of (tt), and (n) is a 
positive integer. 
Let us assume 
(f " + W)” = ) . TT -f <pf (g )7r“ + &c. , 
where 
>p:”(§)=e“+A»v-'+A“5“"-“+&c, 
(P.»g +Bny-"+B® g'— +&C.; 
^^2n + 2_J_ A[f^§^"+&C. 
••• Ai‘i,=AL’^-f^(2.z), A®,=:AL^)+Al^)^(2^^-l); 
or 
Similarly, 
Al*> = ld(2n), A® = 2(^(2« - l)ld(2n}). 
Al^^=l{0(2ji-2)l(^(2n~l)l0(2n})} . . . &c. 
Again, we shall have 
= d'(2oi)p 
f' 
_|_J5(0) .,2n+I 
+ 
+AL'^'(2 
+ 
■RO) o2h I -0(2) 2n-l 
-Dra + i^ ~r -^n+lS 
n—1 -j- A® ^'(2^ 2 , — 2 ~ ' 
-f &c 
+ .. 
Consequently 
+ B^:^2n-1) f”-\-Bi^^d{2n-2) f”-'+ &c. 
B:A=Bi^^-i-6'{2n) ; .-. B»„, = 2^'(2;2) 
Bi*>,=B«+Bi»)^(27z-l)+Al‘>^'(2w-l); 
'. Bi'> =l{0{2n-l)2d'(2oi)) + l()'(2n-l)l()(2n)}. 
Hence we shall have 
+ 2d{2n - l)l6{2n)§^’^-^+ld{2n -2)l()(2n - 1 )2^(2w)^^”-^+ &c. 
+ { 2 ^'( 2 ?^)®^”-'-f( 20 ( 2 ?^-l) 2 ^'( 2 «)+ 2 ^)'( 25 ^-l) 2 ^?( 2 ^^)X^^"-‘^+ &c.}^+ &c. 
AMien ^(t) is a rational and entire function of (cr), 
2^(2?^), 2 (^( 2 ? 2 - 1 ) 2 ( 2 w)) &c.; and 2 ( 1 '( 2 r 2 )&c. 
can always be obtained in finite terms, as manifestly ought to be the case. 
In like manner we shall have 
(s-1-b(^))"=S’-+2«(n)f-= 
JIDCCCLM. M 
