4 02 SCIENCE PROGRESS 



M. Find the value of <f>~ 1 and ^ _1 from the general theorem 

 of operative multiplication. This theorem is 



#!</>! = [Bo + Co 2 + Do' . . .][bo + co + do . . . ] = 



= Bbo + (Cb° + Bc)o 2 + {Db> + 2Cbc + Bd)o" -f 

 + {Eb l + 3^^ + C(2bd + c 2 ) + Be} o 4 + {F6 6 + 4 Eb i c + 

 + 3j D(fctf + ^ 2 ) + 2C(fo + cd) + Bf}o 5 + {G6 S + sFb l c + 

 + 2E(2b i d + 36 V 2 ) -f £>(3& 2 e + 6£cd + c 3 ) + C(2bf + 2ce + d*) + 



Each term of the operator Praises the whole of the operand <f> to 



the appropriate power, and the coefficients of the successive 



powers are then collected. Put $ = </> -1 or </> = $~\ Then the 



coefficients of all the powers of o, except of the first, must be 



zero in the result. We thus obtain a series of equations Bb = i , 



Cb 2 + Be = o, etc. ; from which B, C, D . . . can be deduced if 



b, c, d . . . are given, and vice versa. Similarly, to obtain yjr' 1 , 



i i 



let yjr H operate on o» + co" + do~« . . ., equate the coefficients in 



the result to zero, and solve the series of simple equations thus 



given. The series for <£<£ can also be obtained by the operative 



division of # by </> _1 , and by Maclaurin's theorem, and in other 



ways. Note that the result is more involved if the operand 



contains an absolute term a — as in the value of #o<£o- 



N. Deduce fr l from [0,]". 



We have only to put — i for n in the value of the latter ; but 

 it is not quite easy to obtain this. It is laborious to form 

 <f>i 2 > <f>i 3 , • . . and to observe the law connecting the successive 

 coefficients ; and almost impossible to do so in the case of <f) 

 (which contains an absolute term — see Section V (3) below). 

 But we shall succeed more easily if in the series for &i<j>i just 

 given we put $ x = ^{ l . Then, remembering that <f> n+1 = (£"</> = <£<£", 

 that is, that in this case the two operations are commutative, 

 we can obtain another series for <f><j>" which must be the same 

 as the first. Equating the coefficients, first putting B = b= 1, 

 and assuming that C = nc, we shall obtain D, E, F . . . in 

 succession ; so that 



[o + co 2 -f do* . . .]" = + wi co 2 + (n x d + n 2 2c°)o 3 + 

 + [w 1 e+ $n 2 cd + (n 2 + 6« 3 )c 3 }o 4 -f \n 2 f + n 2 (6ce +3fi? 2 ) 

 + (5«2 + 26n z )c*d + (10M3 + 24?? 4 )c 4 }o 5 + . . ., 



+ 



