78 
ME. W. H. L. EUSSELL ON THE CALCULUS OE SYMBOLS. 
+ S(5(?i-2)2S(%))f-^+2{S(w-4)2(fi(w-2)2S(w))}f-®+&c. 
+ {25 (w)f-'+(2%-2)25(w)+25(w-2)2a'(w))f +& c.}t+&c. ; 
and also 
(§'+J^W)”=r+25(27i)§— ^ 
+ 2(5(2^i-3)25(2w))§^”-®+2{5(2w-6)2(5(2?z-3)25(2w))}§^"-94-&c. 
+ { 25'(2%)§®"-=*+ (25'(2w- 3)25(2%)+ 25(2w- 3)25'(2%))^^"-®+&c. }^r +&C. 
If we put 5('r)=9r^ it is obvious that the three last theorems will give us the ex- 
pansions of 
dx'"' dx^ 
and of 
in terms of x 
dx' 
The same methods of course will apply to all binomials included under the form 
(c*+§^5(‘r))”. I have found that there is no difficulty in calculating the forms of the 
coefficients, beyond the labour expended in performing the finite integrations. 
To determine that part of the expansion of (§'*+§“~'5i(9r) + §““^52('jr) +§““^53(7r)+&c.)" 
which is independent of -r. 
l^et us assume 
(5*+5— S,(!r)+5-=«,(x) + g-%(T) + .,.)"=<)!™(5) + <p™(5).,r+(p<«(g).,r’+,.., 
where 
Then we shall have 
A®,§-+“-^+ A®,g-+“-^+... 
= f”+*+ A^'y”^““‘+ ALV”^“''+ A^V"+*-"+... 
+ 5,(a%)f"+“-'+Ai'>5,(a%— l)§“™+“-"+A®5,(a%— 2) f”+“-^+... 
+ 52(a%)f”+“-"+ A”’52(a%~l) ^“"+“-3 + . . . 
+ 53(a%)f ”+“-"+... 
.-. A^‘i,=AL') + 5,(a%) 
A'^i. = A®+AL')5.(«%-l)+52(«%) 
Al^i, = A'^’+A®5.(«%-2)+AL'>52(a%-l)+53(a%) 
&c. = &c.; 
.-. AW = 25,(a%) 
A^^' = 25i(a% — l)25i(a%) + 252(a%) 
A^®' = 25i(a%— 2)25i(a%— l)25,(a%) 
+ 25i(a% — 2)252 (c 4%)+ 252(a% — l)25i(a??)+ 253(a%) ; 
and consequently the part of 
(f +r ’^.w+r •• ■)”. 
