(unless in thjeext^eptedcas^ of y asp^ J^at therefore by the 



ordinary ^ule, ^= "Zpq, z=p^ + q\'^y^p'^^'g^\^&[\cQ 

 p and q are prime to each other, and ,,•__, 



4- J — s« 



— = 1 + 



6J 21 fcilDG^ i9r#tol Dili /fj ~-^j 2 ?.b990X9 -lo aliJfjpQ s« woK 



•I9lfi9i§ ^^IlBiJTf!j^a^2^kMaff^^-«i»i|im.^-r(^^^d tej-jngy/ij 



jiavui 9(b iu>rfy/,,rn'^9J-hini prlJ.gc 'jpl an ^tiMO I'jb ilJ 



fe#W|Wfi^lS^i^ffl«t44»aj§lB.fyyfe +j/) fQR 



^_/x+yy g^)Tfo-(2yaWT!ft?f . > r|- to — li/'"-^ + (g??/'»- ' -^z +y) 



and as ;r + 2/ is even whilst m and »z— 2 are odd, there would 

 still remain a fractional ^. Further, if 2; +3/ = 4/, z +y is of 

 the form 4j + 2, so that one term will have \ and the other |, 

 for a remainder. 



Hence, as p^ q-'^ — a quantity -^ by (2; +j/) -r- {z—y) to the 

 first power only, it follows that {z-^y) -^ {z—y), or their pro- 

 dmct z^—y'^, must be a square. Putting therefore z = r"^ -{■ s% 



^^Sfnt^orrrJ^tinU .(_-ioJ:i(.l«!lJ;..ni- o:i.i(> ■ . li ii-x/a ,3i.;fu 'jijon e '^d 



y^ut](U, g^are real and prime ta each other, so ar6"r ihtiTsX 

 .•^as we cannot suppose p^ — 5'^= (r^ + s^)~'" without introdu- 

 cing the imaginary sign of \^ — 1 iii the factorial result, 



He»ce KiiIdcJe9 ^Ilxj't 02 9L iqiR X 



jDjihsfi 9ri4o2_9aix)BneK|aJm ^^^ _ , hog ^rfj-fiio 



opoilBi9bianoD eiriT .zlBmiaa 'loiialai arij lo bne npra lo boot 

 isoilqat' ' 2"w -i- *I ^^ora siuoo'iq oi aJqfnaJJB ^nfiffi 0} bgl scd 



'"-r/sh^?''** s ^i!?0''fo yongioftab B bb ^bioB ofiodq^odq lo 



889T;^o*iq ai ^ ^ :_^u{S!fL|uorii^2.Gjq9&9'iq i9V9 Jg-orn ,Jn9b 



ilaidw fno4* ^ L> . > t ' - ■ ... .'' PT, Jnariiav y^^Tx?! jis<jtiii|to|^Blo 



,89nod nggd 9VBd J: 1 • 2 .. 2 ? .!9ijd and bioB oi'to^lq^oiiq 



9(tJ oj 7U9*9iib ii vlqqn « 1 . 2 . . ? gvcrf aniio wnii^olqaia lo 



Setting out from either end, the extreme quantities are 

 equal, and have the same coefficient, unity, but the upper co- 

 efficient afterwards more and more exceeds the corresponding 

 under one, for at the hh. term they equal their immediate pre- 

 decessors multiplied by arlj \(d biiirjinirmfaoD ^ 



J ,C4^8I t<iCioViO .081 .oK .VS.^ JoV .e .d .T^^ifv .\sin 



