292 SCIENCE PROGRESS 



to observe how the functions of geometrical progressions in the former become 

 degraded to binomial coefficients in the latter. 



If n= - i, G r = -b~ r and K rs =b~ r ; and <£ -1 gives the same series as that of 

 3' 5, except that O and c, d, e, . . . are now all divided by b. If n -* oo and \ b \ <i, 

 G r =i/{i -b r ) and K rs = bs/(i-b r )(i-b s ). 



Examples, 

 [bo + cO 2 + do 3 . . . ] 3 = b 3 + {i + b + b 2 }b 2 c0 2 + {(i + b 2 + b*)b*d + 

 + (i+b + b 2 )2b 2 c 2 }O s + etc. 



[bO + cO*] n = Bo + B I ^ C l O* + 2£ I ^^-*?0 3 + etc. (B^b n ) 4-3 



b-i b b-i b 2 -1 b 2 v ^ ° 



This last example is the solution of the difference-equation x n =bx n -+cx\-\ 

 which denotes the time-to-time variation of malaria in a locality — a problem 

 which led to part of the studies outlined here (see my Prevention of Malaria 

 in Mauritius, Waterlow & Sons, 1908, and Prevention of Malaria, Murray, 191 1). 



5*0. The analogy between the algebraic and the operative multinomial 

 theorems, that is between the expansions of (cf>) n and of [<p] n , or <£ n , respectively, 

 is best shown by evolving both by the same method — which requires only a know- 

 ledge of Pascal's Arithmetical Triangle, according to which the binomial coefficient 

 n r is the sum of the first n — r+ 1 terms of the rth order of figurate numbers. Let 



(<t>) n = (0 + c0 2 + d0 3 . . . ) n = O v + c n O n+1 + d n O n+2 + etc. ; 

 0" =3 [o + cO 2 + do 3 . . . ] n = O + C n 2 + D n O s + etc. 



First, to find c n , d n , e„, . . . , we have (<£)" = (<£) n-1 . <fi ; and multiplying alge- 

 braically the two series (<£)"" * and <£, and equating their coefficients with those of 

 (<£)", we obtain 



c n = c+c n -i; d n = d+cc n -i + d n -\; e n = e + dc n -i + cd n -i + e n -i ; etc. 



Now on adding, for example, e+e 2 + . . . e n .i + e n , we find that a number of terms 

 cancel out, leaving, for example, 



e n = ne + d(c+ Ci . . . c n . -2 + <r„-i) + e(d+ da . . . d n - 2 + dit-i) ; 

 with similar results for e„, d„,f„,g n , etc. Hence in succession, 

 c n — nc 



d n = nd+c(c+C2 ■ . ■ c n -i) = nd+n 2 c 2 

 e n = ne + n$cd+ c(d+ di . . . d n -i) = ne + n 2 cd+(nid+n3C 2 )c ; etc. 



Hence (</>)*=0" + ncO n+l + {nd + « 2 ^}O n+2 + {ne + 2mcd + n t c*)O n+i + 

 -f- {n/+ n 2 {2ce + d*) + yt%c 2 d+ n i c i }O n+i + etc. 51 



This is the algebraic multinomial theorem. Similarly to find the operative 

 multinomial formula we equate the coefficients of <£ n with those of <£ n-1 <£, since 



O + CnO 2 + D n O*. . . = (O + C1O 2 + d x 3 ...) + C_i(0 2 + <r 9 0»+ d^O* . ..) + 

 + £>n-i(0* + c 3 O i + d 3 & . . . )+etc = + (c 1 +C n - 1 )+(d 1 +c-2C n -i+D H -i)0 3 + 

 + (ei + d2C H - 1 + c 3 D H - 1 + En-JO* + etc. ; 



so that the coefficients are the same as those of (<£)* except that subscripts are 

 attached to the small letters. Finding 2C, S^», ... in succession as before, 

 we obtain C H =nc, D n = nd-\- n^c^, etc. ; and have 



[<£] n = O + nciO 2 + {ndi + « 2 <V2}0 3 + {ne x + n<&c\di + c 3 di) + ntCiCtf*}0* + 

 + {nf\ + n?,{ciei + d\d 3 + ae{) + nz{c\Cid z + c\C^d% + csadi) + n^ciCid) O 6 + etc. 5*2 



