298 SCIENCE PROGRESS 



[^"]«=[(*-i)f]^0]»[(*- I V 1 ]- 1 = *« . ,0i»] 



.n n, , n,, 



[(^"Nfo-Ki + offlo-'] 



Evaluate . I by several methods. 



Show that [^]"=f[|fl"f- 1 =|- 1 [|^ and that [%fH=*[WN=MxW 

 [#a + <*o)]" = [^][> + ^]»[a + fo] 



[a + ^O^ + fO- 9 . ..J»=l/[«- 1 -«- 9 ^0-(a-«r- < i-»^)0*...] ,, 0- 1 



[(a + £o) m ]" = [O m ][a + bO m ] n f y5" 



[O - hg{i '+ e°)] n = /^]"«° = o - /^ i + »* °) 



[QW+logo)Jt _ Ql/(l+nlocro) 

 [ e o-l]n = [e o] [eo _ l]V ^. 



[I + %0]" = [€°][/^I + o)]»/^o 



[/a«[jr + o]/3»- 1 0]"=[^~t^;J"=[o]", where « may be fractional, and 



= fan[ir + 0]«tan~ l O = tan{mr+tan-^0) = iantm +° 



I - O . tan mr 



[tan(nl 4 + tan-*0)]* = [ I + °'Y- tannn^ + O 



n Li-oJ ~i-0./anmr/4 



[sin[n + Ojsin ~ l O]" = [ - O]" = sin mr . *Ji - O 2 + cos mr . O 

 [cos{rr + cos _1 0)]" = [ - O]" = cos nrr . O - sin nrr . s fl - O* 



J L J cot tin I* + 



[wi(ir/2 + sin~' L 0)'Y = [ >Ji-0*] n =sin mr/2 . Vi-O* + cos mr/2 . O 

 [cos{n/2 + cos - x O)]» = [ - Ji - O*]- = cos mr/2 . O - sin mr/2 . jT^O* 

 [cosO + i sinO]- 1 = cos- 1 $(0 + O" 1 )— which is unreal. Hence 

 [[cosO + 1 sin o][> + O][cos~mo + O' 1 )]]" = [ - £(0 + O" 1 ) ± J(0 - O" 1 )]" - 



= [-0]"or[-0- 1 ] n 

 *»<:<?j mr . O + 1 sin mr . O -1 ; or =cos mr . O -1 + 1 sin mr . O, 



Derive numerous periodic operations by using the CiC' 1 method when $ - - O. 



Study the iteration of circles, ellipses, and hyperbolic functions. 



If O" is an operator which raises <f> to the «th operative power (see l'o), show thai 



-O -1 is a versor which turns <£ through a right angle clockwise ; and explain 



how it is that Q° converts <f> into the midaxis, O. 



1 



References to the Writer's Papers 



1. " Verb-Functions, with Notes on the Solution of Equations by Operative 

 Division" {Proc. Roy. Irish Acad., xxv., A, 1905). 



2. "A Method of Solving Algebraic Equations" {Nature, Vol. 78, October 29, 

 1908). 



3. Ibid. {Nature, Vol. 79, February 4, 1909). 



4. 5, 6. " The Solution of Equations by Operative Division " (Science PRO- 

 GRESS, October 1915, and January and April 1916). 



