LÉON AUBRY. DÉMONSTRATION I>l THÉORÈME DE BACHET. 65 



En remarquant que 



^« 2 ^= ^n 1 -(- A„ I N,, ], 



on en déduit 



— \,i 2 1 + V„ oo -+- X„ -1 I = — A„_,N„_i, 

 X/; iX/i 1 -h Y„_2 Y„- I H- I = !\„ 1 f A„-.,X„ -1 -+- l>„ lï/j-i + i], 

 X/i-2Y,j 1 — \„ 2 X„- 1 -!- o. I = \„ 1 [ A„ 1 Y„ 1 — B,j-_iX„_]], 

 X„ 20 H- Y„-2 I — Y„_i I = B„ iN„^.,, 



et, par suite, à cause du théorème I, en remarquant que m — n = k = i, 



p = Z = i, /■ = U = o, s = X„_,, q = Y„_i 



(21) X2_,--Y?,_,-hi = N, ,X„ 2; 



puis, en posant 



(i->.) — A,;_, =z a„ 2, 



(•23) A„--i X„-i-t- I5„ , Y„ 1^-1= c„ -2. 



(24) A„-,\„_i — B„_iX,i_i = d,i-i, 



(•ij) h,i-i = bn—u 



il vient, 



(9.6) \„_2 = «',-2 + f^îl-i -+- <"« 2 -t- dfi^^_, 



(27) — «„-2X,; '-i- 0„ -il/iî-hCn 2=^ I ^/i-2? 



(•28) Cn^X/j 2 Of/, -2 1 '( 2-T-'^'/i 2= ^H lN/;~2, 



(■29) dii-î^n 1+ C„ 2Y„ 2— b„ 2=Y„-i\„__2> 



(3o) 6/1 2X,, 2-!- ««-2 Y„-2 4- fi?„ 2 = O N/i-2- 



Soit maintenant 



( 3 1 ) N/ = a| — i/ ^- c'j 4- rff . 



( 32 ) N/+1 = a;v 1 -4- />>/+ 1 -+- cf+ 1 + df^ 1 , 



(33; X;4_Y|-f-i=N,N,+„ 



(34) — a,\i-r- />/Y,-^ Q= «,.^iN/, 



(33 j Ci\/—diYi-\-a-= c/+i\/, 



(36) df\,-h c/ Y,- 6, = rf/+i N,-, 



( 3; ) />/ X,- + ai Y, + (/,• = />,+i N,-. 



Si nous remarquons que 



X,- I = \i-T- A,X,-, Y,_i = Y/ +- B/'S,, 



on a 



— a,\i 1 -^- ^, Y,-^i -i- Ci = X/[— rt,- A;4- ^/B, -f- <:/,>, ], 



C,X/ 1— rf,Y/„,-t- C/,= N;[ C,-Ai— f/,B;-t- C/+1 ], 



f/, \/ , -T- c/Y,- I— /;/= N/[ d,\,-i- c,P>,H- «^/,+i |, 



/;/\/ 1 + a,\i i-T- di= N,[ ù,\i-i- «,B/ - /^,-+i], 



