p EY — Real and Complex Numbers as Adjectives or Operators. 29 



ch? ,. . . e 



= limit of 



p {log [r + dr) + ((« + <« ±2iir)} _ g p {logr + i(fl ± 2/nr)} 



-- - lumu u± - g logr + *-+i(8K?«±2* 1 r) _ g log »• + i (9 ± 2/«r) 



p {log »■ + «(« ±2^)} | c io{log(|- + rfr)-log)' + irffl} _ 1 J 



= limit of 



limit of 



log »■ + i (B ± 2iv)i log (r + dr) -log r + idd _ i' 

 £ (6 J- i 



£P Lp{io g (i+ *)+■*») _ i 



|e 1 °s( 1 +7-) + ^ 9 - l' 



,!»(* + ««) 



i) ^'h- 



/(fr 



I j- + p f — + if/0 ) - 1 

 = limit of L- . = limit of 



(v +m ) 



(dr , \ civ 



• ■" 1 + — + tdfl-l 



r 



= JjaP" 1 . 



Finally, as an example of a way in which this mathematical game may 

 be visibly played — for of course it may generally be played only on paper 

 when the units a', (5' are of any different kind whatsoever from a, j3 — take 

 4 measuring vessels ABA'B' arranged in the cyclical order AA'BB', and 

 pour any quantities of water into them. We lay down that the position of 

 the game at any time is given by stating the difference of the quantities of 

 water in A and B and stating which is in excess, and making a like state- 

 ment for A and B'. Then what is meant by a number 2 or + 2 operating 

 on the quantities, is the same as doubling the quantity in every vessel ; 

 2i means that the quantity in every vessel is to be doubled and transferred 

 to the vessel next to each, say on the left ; 3t 2 or - 3 means that the 

 quantities are trebled and transferred to the vessels second in order on the 

 left or to each opposite vessel, 4i 3 or - 4i means that the quantities are 

 quadrupled and transferred to the third vessel on the left or the first on 

 the right, 2 + 3-i means that the original quantities are to be doubled, and 

 also that they are to be trebled and transferred to the vessels on the left, 

 and so on. Now we have proved that the final position of the game after 

 any number of such orders have been carried out depends only on the original 

 position, that is on the excesses and the vessels containing the excesses, 

 and indeed that this is so is obvious ; that any order implied by x + it/ 

 can be carried out by a repetition of any one of n orders each repeated n 

 times, and we have shown how to find these orders or operations. Some 

 problems are obvious, thus the order + 1 may be effected by repeating 

 twice the order + 1 or the order - 1, or by repeating 4 times any one of 

 the orders + 1. - 1, + i, - i and by the repetition 4 times of no other order. 



