ESSAYS 297 



The quotient being the required operation ££f -1 . Here each term of the quotient 

 operates on the divisor instead of being multiplied into it as in algebraic division. 

 Now, equating coefficients, inserting values in the quotient, and then putting b n for 

 b in the functions already in the quotient, we obtain 



^*m\bO + c + do- 1 + eO~ i ...] n =-Bo + GxC + G^djB.O' 1 + {GtPe/B* - 

 -(L7 a £/i?-6^7# 2 )^//(^ 3 -i)}0- a + etc. (B = b; see 4-1). 10-3 



Similarly, but not by the above method, 

 [O + c + do- 1 . . .] n = O - c- dO^ 1 - (e + cd)0~* + (/+ d* + 2ce + c*d)0-* + etc. 

 We can also obtain both series by the process of sections 3*0 and 4'o. But we 

 can also derive the value of \j/ n from that of <£* and vice versa by employing 

 another C^C" 1 transformation in which £=o~ 1 = £- 1 . Then 



[bo+c + do- 1 . . .]"=[o- 1 ][(*o- 1 +f + do. . . )- 1 ] n [o- 1 ] 

 =[o- 1 ][^- 1 o-^-Vo s -(^-V-^-3^)o 3 ...] n [o- 1 ]. 



This last operation can be iterated by 4:1, giving b~ n O — G\, £~ 2n <:0 2 -etc. ; so that 

 finally ty H obtains the value shown above. 10*4 



(4) If in the first example in 3*6 we put c=zjn and suppose n becomes infinite, 

 we have 



f"o + -O i J =0 + zO t + z*0 3 + z 3 O i + . . . 



= 7-F6 = Lr^ J ■ io 5 



if the subject of the operations is numerically less than unity. This result appears 

 to be the homologue of the algebraic exponential theorem, and I therefore denote 

 0/(i-0) by 77. Operative logarithms are also possible and may be denoted by 

 the abbreviation oplog. 



110. Conclusion. This paper does not pretend to deal with iteration in 

 general, including vortices, etc. It only outlines a chapter in operative algebra ; 

 but such a chapter is necessary for the application of operative algebra to 

 the Calculus, which deals with the iteration, not of constant operations, but of 

 operations which vary at each step. I hope to publish some day my work on this 

 and some other developments of the present theme. 



12*0. Examples. 



[a + ^0]" = | + -^7j[^0] n | O- -^J = 67 1 a-f-^"0 (compare 3*1 and 4*1) 



[0 , -j0 5 ]"= x /2 + [0-2 N /20 1 -40 3 -^V20 4 -|0 5 ](0- *J2) 6-3 



[a - s/a-0] H =a- 1 + [O + Jo* + AO 3 + tV 3 4 . . .]"(0 - a + 1) 6'3 



[O-C0SO] n =7r/2 + [O + sinO] n (O -tt/2) 6*3 



[a + ^0-]-[0-][ a -^J[0-] 



[Va +p + b(O m -P)] n = [O m -p] ~ x |> + bo]»[O m -p] = m yGia +p + b»(Q™ -p) 

 [ (g + bVo +py* -p]" = [O m -p][a + bO] n [O m -pi = ( Gia + b». Vo + p)™ -p 



[Sb~+pf -OJ J / P •> J nP-{n-i)0* -y 



L " T^j|_ l _ Qlp j L« F\ LV2-0V/J V I - »(0* -$)\p\ 



«f=[0/»]W[»0] ; 0r=[O-»][i + O]»[O-i] 



W=[0- ^\bO\\{b- i),]-» ■ or = [o-i][i±?]"[o-i] 



-*» . i,»i. 4-3 



