4Q0 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779- 



The quantities + p" and + 2aVj^/)^ denote + p-, and + la^b'^p'^, \{"~ is a 



whole number, otherwise they denote — p" and — la'^b'^p' respectively. 



5. (1st) Let ar = fi^/j+ b!^p"-\ and n an odd number; then will x" — 

 nabpx"- + n . '^^ a-byx-* _ „ . !iz_\ ^ a'byx"-^ + « • ^- • ~ ■ '^ 

 (("bYx"-^ — &c. = a"p + by-K 



5. (2d) Let X = al/p + b^P"''\ and 7i an even number; then will x" — 

 7iabpx"- ' + «•—- a^y-x" -4 _ n . -— . -— a^byx'-f' + n . — - . _-^ . 2-1 



n n n n n n 



a^byx"-^ — &c. = a"/; + 2a^by + ^"yb"-'. It will be + In^by if n = Ar + 



n « « ^ 



2; but — la'-b^p- if tz = 4/-. 



6. (1st) Let x= al/p ■\- b''^p''-^, and w an odd number, which has not the 

 number 3 for a divisor; then will x" — naHipx'-i -f n . — - a^byx"-^ — n . 



. —J- a^°byx'-'i + n . — — . — — . — — . — — . — — a^^byx"-^ — &c. (to 

 m terms, where m is the number either equal to, or the least greater than 



3/ /- \ ' * 2 3 -f^ '2' .2 3 4 .0 



g— 15 n - 21 



4,2,0 2 I 1 vv ""-'^ n— 11 72-13 n-15 n— 17 « — ig 6;i 3 ~~j~ ■ 1 



n — 27 



n — 11 n— 13 n— 15 k— 17 n— 19 « — 21 ?;— 23 n — 25 o,. 4 ~~ . „ , 

 "• ^- • -i~ • — - • — - • -§- • -7- • -^ • — - «^ A + &C.) = A 



■= cfp-\- by-^. 



Let n be an odd number divisible by 3, then will the above-mentioned quan- 



« 2n 2n zn n n 



tity = A = a"/j + by-^ + Sa^b'^p^ ' ' + 3aJby. 



6. (2d) Let n be an even number, not divisible by 3 ; then will or"— na^^^a?"-3 

 _,. „ . ^la^byx"-^ - "-^^ • ~o^byx''-'^^n:^ . 'L-J2 . 'L-Jl a^yx"- " 



_„ "-' ' ^i£ !izi3 ^If a"'^'Va.'"-'5 4-n — - — — — — '-i:il^ ^IZ 

 * 2'3'4*5 " ~'"*2*3"4'5'6" 



-3 



a'^6*/>V- '* — &c. to m terms as before — by {2x^ -f ^n . ~ a'bpx^ + 1 

 n34 ■* '2' 234 56 



k3 "z"' ■ 1 



^3r* -4_ _ V ■» ' " ZI-L: 1__ " '" "— '" «-_f^ "—f /,8/,4h4 2 , 



/>* + 27 X "• "T • ~ • 4 -"5" •""6~''~7 -^^ ^* + 



&C.) = A = a"/) — 67,"-'. 



