262 
ME. W. H. L. EIJSSELL ON THE CALCHLHS OF SYMBOLS. 
where aj, /3i, y„ . . . v, (3^, 72) • • • &c. are all the whole numbers which satisfy the 
equation 
«5-f/3+y+ • -\-v=r. 
In the preceding investigation we have used as an abbreviation for 2 ^ — m' 
where is the mth function derived from In the following investigation, it is 
proper to remark that ^),(7r), &^{Tr ) ... are any rational and entire functions of (tt) 
whatever. 
To find an expression for the general term of the multinomial theorem, by which 
(f+r'U^)+r%H+r%H+ • • • )” 
is expanded in powers of (§). 
Let us assume 
(f-+f-v,(x)+f-x(^)+ . . . )-=r +r'VS"’(’r)+e—’pf’w+r‘V?'W+ • • • 
Then multiplying internally by the factor 
and equating coefficients of like powers of (§), we have the following series of equations : — 
(tt + a) = 
^^”+'^(7r) — a) = 0,(7r)p)^”^(7r + a — 1) + ^2(77), 
_ (p(»)(T+ a) = 0,{7r)(pi^\7r-^a—l ) + ^2(7r)?)^”)(7r + a— 2) + ^3(x), 
d d 
and thus we proceed: hence we have, putting the symbol 
®^”^(7r) = £““5^^ n^iTT, 
^("V=g“*5^ n^i(‘r)g“*; n^,(T)4-2”“^ Yl&J^‘7r) 
n^i('r)g"*^ n^i'rd-g““^i- T[o^{y)z~T7r n4(7r)+g““^ ro 
n^i(7r)g“^ n^i('>r)g“^ ni^i('r)g~^ n^i(7r) 
-}-g~“5^- Y[6iprz~d^ 
-j”£ *d7r n^j^TT^g liTT II(?3^7r)-|~2 "drr n^g^TT^g II^i^TT^ -j~ 2 *rf7r II^g'^S^^dTr II^2^“1''^~°'‘^’' II^4(‘^)’ 
We easily see that the general term may be expressed thus: construct the formula 
£““^ II^„(7r)g“®^ Tld |^(7c)^~^di YlSj^‘7r)z~'^di . . . n9^(7r), 
and give to a, 7, ... v all the values which satisfy the equation 
a-\-h-\-c -\- . . . +e=r, 
where an—r is the index of (^). Then the sum of all the terms so formed will be the 
required result. 
To determine the product of the factors 
(f+%nW)(f+%n-i(’r)) . • .(f+%(7r)) . . .(?+7„-,(7r)Xf+%„7r), 
whence %i(7r), X 2 {'^)^ Xsi'^) emanate from each other after a given law. 
