ME. W. H. L. EUSSELL ON THE CALCULUS OF SYMBOLS. 
263 
Let 
• • • (f +%’*■) • • • . . 
Then multiplying internally and externally by f+x;„+i(7r), 
+f'”^'{^2"^(7r+l}+9i”Y7r+l)>i„+i(x+2nd-l) 
+ 1 W +%»+ + 2%+ l);^„+,(7r) } . 
TT en p.p 
?)^”+ ' V — (p^”>(7r + 1) = ,(7r + 2?^+ 2 ) +>^+ ,7r, 
whence d / d d\ 
cp^"\Tr) = z”dn 2 (g<’*+ i(7r), 
and also 
+ ’^(tt) — <p«(7r + 1 ) = (p[”^(7r + !)>;;„+ i(7r + 2 to + 1 ) 
+ ^^"’W%„+I(7^)+%„+l(7^'+2?^+l);^^^l(7^); 
+ £”5S2£“^"+*^*^;:^S»+iW£”^ %„+i(7r) 
+ £V.2£-^"+’>^- %„+i(7r) {£'""+ '^^ %„+,(7r)} ; 
and in like manner we find the values of the succeeding symbolical coefficients. 
I now come to the form of the binomial theorem which is reciprocal to that pre- 
viously investigated. 
To expand .-ry in powers of (t). 
Let us assume 
= ' + ^2»-2(f) '+..., 
we know that 
) =K§)' <+ ^ (f ^ ^ (f +•• • 
Hence, multiplying internally and externally by 5r^-l-^(^)T, we shall have 
+®2»-2(f) •’*^”+?>2«-3(f) • + . . . 
+ {«(e)T“+ 2« I) «(f >“-■ + 2« . (f I) ’«(f 
+fe.-.(f){«(f)- ’r’-+(2«-l)(f I) , . .|<r 
+9»-a(e){^f)-^“-’+(2«-2)(f !)%).»“-+ . . j’T 
-f&c. 
+ |?>2«-2(f ) • ^ '^2«-2(g') 1’*^”“^ 
2 N 2 
