TBANSACTIONS OF SECTION A. 601 



the fundamental conception, and to show that the system may be employed with 

 advantage in determining both critical and criticoidal forms of "all degrees. 



In the usual expansion of the binomial (.r + ?/)" introduce .z" and if and chan"-e- 

 indices into sutEces: we thus obtain : — ■ 



, n(n — l) 7i(n — \) 



■'n!/o + nx„.,y^ + -A__/^„_„y^ + . . . + _^j__'.r,y„_„ + n,r,y,._, + .v^,„ 



in which x^, a\, ,r,, . . . .r„ and y^, y^ y.... y,, may be regarded as independent 

 avbitraries, and may therefore be replaced by any functions or forms we please. 

 I^et this expression be represented for shortness by the binomial (.i" + y)„ ; then tho 

 symbols .i- and y may be called umbra, and the symbols (.r + _?/)„, x,., y,., potencci. 

 An umbra is a mere recipient of suffices, being otherwise uninterpretable ; but in. 

 the particular case av = .i'", which will often occur, .r may be called a radiv. 

 Jtadices are not necessarily algebraical fimctions ; they may represent operations as 

 well as quantities, subject only to the index law .r"'y" = ,i"'+". If ,%■ be an umbra 

 and y a radix, the development of (.r + y)n w'ill be 



n(n-l) , n(n — l) „ „ 



■'" + «•'■"-!?/+ \.2 -'— ■^y+ • • • + — jTy-J'-'y +"-''iJ'"~ +-'"o,'/". 



The denumerate form 



••JVyo + '^«-iy.+-'— 2^3+ • • • +-''.yn-.> + -riy„-,+.roJ/„, 

 in which .v and y are both umbral, may be obtained directly from the expansion of 

 (.i+?/)„ by simply suppressing the factors containing n: this form may be re- 

 presented, in accordance with the quantical notation, by (.r + yj„. If .v + y be 

 pen umbral, that is to say, if one of the symbols, .r, be an umbra, and the other, y, 

 .1 radix, then (.) +yj „ will represent 



Writing w,. for - — '"' \ we have (x + y)„ = (.v + mr) „, provided that, in tlia 



(n — r)lr'. 



development of the dexter, we interpret Hq by 1. In like manner we have 



ix-y)n = {x-'>tyj „=.T„yQ-nyV„_iy^ + 7i.,x„_.jy^-&c.; where the signs connecting 

 tlie monomials are + and — alternately, and the general or rth term is 



(-)'+'*'r-rl-«-r+i2/r-i. 



Two peculiarities in these forms deserve notice. One is that in the potence re- 

 presentation {r + nyj „, the n inside the brackets is umbral, and the n outside is 

 quasi-numerical. The other is that in developing such a potence as (.i' + y)„,+„, or 

 its equivalent {x ±m + n .yj ,„+„, the factors {m + n)j, (?« + n).-^, Sec, are not to be 



expanded as potences ; for {m + n),. is simplv what n^ or — becomes when 



(n - r) ! r ! 

 for n we substitute ??i + 71 ; that is to say 



/ % (m + n) ! 



(m + n — r) ! »• ! 



There is no difficulty in extending the notation to any number of symbols. 

 Tlius 



(.r + y + s),. = (r + y),. So + n^(x + y^-^z^ + n.^{x + y)„_,=, + &c., 



and the full development is obtained by expanding the binomials. 

 Similarly 



(.r + 2/ + =) „ = (.1- + y) „ z^ + {x + y) „_^z, + (.r + y) „_,3, + . . . (.i- + y) „s„_, 

 + (.r + y),=„_j + (.r + y\3„ 

 = -i'nyo=o + -'"n-i2/i=o + -^»-2yjSo+ • • . +.r,,y„_2S(, + .riJ/„_i2o + .r,,y„Sn 

 + •^"..-1^0=1 +-'*'n-22/iSi + -2„-3y2=i+ • • • +a:oy„_3S, + x-iy„_2:, +;ivy„_i3j 

 + ^n-i>Jffii + ^'n-3yi^i + x„_^y,Z:,+ . . . +x,y„_^Z3 + x,y„_jrj + a:oy„_j3, 



