I'ROF. A. C. DIXON ON STITRM-LIOUVII.LK II \1;M<)NIC KXI'ANsloN^ 

 < 'han<,nn<,' into \(f, or u into /-. .,r U>th, we him- similarly 

 *(6,a) ! o(a,b) = -<l> (/>,)+ U (/-.") 



1 1 '.i 





' " > v < '- 



% a) U (r, 



- r) * (,-, ) V (,', 



These are ezpntliona for the errors committed when n, r, U, V are taken as the 

 values of 0, \ff, 4>, ^. 



Ijet /u denote the upper Itmindury of the ratios of 



] %/X i^ (*!. - r )- " (-''I. - r o)} | , | <J> (^i, A)-U (o-i, a-o) ! , 

 | ^ (.* .,-)- r (x,, a-,) | , I X' w {^ (a-,, a-.)- V (.r,, r )} | 



tr> c'xp (.r, .r,,) for values of ./, .', such that 



a 2 r,, < .r, 2 I. 



Let the symlx)! // denote equality in order of magnitude so that P//Q means that 

 the ratios !'/(,), Q/P are both limited. 



Then in the expressions for the four en-ore, since h^.r^a, 



\ }J (.r,h), *(.,-,/,), U (./-,/>), X- ia V(.r,/>) are all at most //exp (/>-.r), 



while 



X 1!l ^ (.'-,<-/), \js(.r,rt), *(./', a), X' 1 -''*!' (., a) are all at most // (l +/x) exp a (.1- rt). 



Hence such a product as * (.r, a) U (x, /*) or \^ (^, a) '< (r, 1>) is // ( I +M) exp a (/>'<) 



at most, ,-ind 



f j( P - r) * (a?, ) U (r, />) + X ( 1 - I) v, (.,-, a) (.r, /v) [ ./ 



.' I V r/ 



is at most // w (I +/x) X P (/') where 



r* 



Also |*|<2z i sui)jK)sed to lx- finite, and therefore 



Jo 



r* 1 4- 



rr<f>(.c,<t) n (,i-,l>) </.r is at most // - 

 .u \ 



3 H 2 



(/> a) 



