290 SCIENCE PROGRESS 



expression of a " midaxial root " of the multinomial, as given by operative division, 

 and other terms (7"o). 



If, however, the multinomial has no absolute term, its iteration-formula can be 

 obtained more easily by several methods. 



The formula of 2'4 gives the operative product of two multinomials *<£, 

 without absolute terms. By interchanging the capital and small letters in this 

 formula we can write out the value of <£$. Now suppose that $ is the given 

 multinomial of which we require the iteration formula, and assume that * =m <£", 

 its coefficients B y C, D, . . . being required. Then since $ n $ = <£<£", the expansion 

 of *# is equivalent to that of </><£ ; and by equating the coefficients of the same 

 powers of O in both, we can determine the required capital coefficients in 

 succession. 



First assume that b= I. Then obviously B =\ also ; and we have the series 

 of equations 



C+c=c+C 2Cc=2cC zDc-rC{2d+c l ) = 2>dC+c{2D+0\ etc. 



The first two equations are identities, but from the third 



cD-Cd=cC*-Cc*; 



and so on. Now it is easy to see by commencing a few substitutions that C= nc\ 

 so that D — nd^-n{n- i)c*, and E, F, . . . can be similarly found. Hence, writing 

 n, «a, n», »<,... for the successive "binomial coefficients," 



(p n = [0 + cO i + dO s ...] n = + ncO z +{nd+2n i c i }O i +{ne + ^n- i cd-{-(ni + 6nt)c t }0*+ 

 + {»/+ 3n*(2Ct> + d 2 ) + (5n< i + 26n i y i d+(ion l + 24ni)c i }O i + {ng + yn^cf+de) + 

 + 7("a + S n *yd* + 4(2«a + 9 n ») c * e + (n 7 + 7i» s + i$4m)c*d + (Sn» + 86*< + 

 + I20« i y s }0 8 + {nh + 4n^2cg+2df+ e % )+ i2(» a + 4«») + c*f+ 4(5«a+ 22»»)cde + 

 + 3(»a + $n % )d 3 + (3» a + i63« a + ^on^d 1 + 4(« a + 3 ln » + 6on*)c 3 e + (94«» + 

 + 8o8«4+io44«»)^+4(«» + 43«4+ i89«i+ i8o« 8 y , }o 7 + etc. 3 '3 



This series has been given by me previously (5, p. 402). To establish it for all 

 real values of n we easily show by the use of 2*4 that <fi m (j) n = <f> m + n ; and the 

 remainder of the proof follows that of Euler for the Binomial Theorem. Also 

 [0 m ] n = cj) mn . These results show that permanence of form exists for operative 

 involution as it exists for algebraic involution. For verifications, we find that 

 when n = i,(f) 1 = 4>, and that when n — o, 4>' = Q, its correct value (see 6, p. 593, and 

 1). Also the homographic result of 23 is verified by the result 



t-°-T=[o + O s + 5 • • ]" = + »0 , + » J 0* . . . «* ° . 34 



i-OJ 1 -nO 



If «= - 1, we obtain the invert of <f> — that is, (f)' 1 ; namely 



,-p-is=[0 + cO % + do 3 . . .]- l = - cO x - {d - 2C*}0 3 - {e - yd + 5<r»}0* - {/- 



- 3(2^ + d*) + 2 1 c*d - 1 4c* } O 5 - {g - 7(cf+ de) + 2%{cd* + c*e) - 84^ + 42^ } O 8 - 



- [h - 4(2cg+ 2df+ e*) + 12(3^/ + 6cde + d*) - fo^e'd 1 + 2c*e) + 330^ - 

 -i32f*}0 T -etc. 3 - 5 



This is one of the inverts obtained by operative division (see 4, 5, 6, especially 4, 

 p. 230) and can always be used for calculating roots of equations, since if <px =y, 

 x = <f>~ y _y ; but we must often shift the origin to a point near the required root 

 (4, p. 231). This particular series is also known from Lagrange's and Burmann's 

 Theorems. It is not the complete invert of <f>. 



