PHILOSOPHICAL TRANSACTIONS. 

 Table ii. Table hi. 



Table iv. 



VOL. LXXXI.] 

 Ta ble I. 



Samofl + I + lSumofi-i + ^„-iSamof^„+i+iSumof3i + ,^^ 

 + &c. ad inf. + &c. ad inf. 



79 



+ &c. ad inf. -|- &c. ad inf. 



Sum 



A 



B 

 C 

 D 

 E 

 F 

 G 

 H 

 I 

 K 

 L 

 M 

 N 

 O 

 P 

 Q 

 u 

 s 



T 

 V 

 W 

 X = 



y = 

 z = 



a'= 

 b'= 

 c' = 

 d'= 



e'= 

 f'= 

 g'= 



k'= 

 l' = 



m'= 

 v'= 

 o'= 

 p'= 



: .644-934066848 

 : .202056903159 



: .082323233711 



: .036927755107 

 : .017343061984 

 : .008349277387 

 : .004077356198 

 .002008392826 

 .000994575128 

 .000494188604 

 .OOO226O86553 

 .000122713347 

 .000061248135 

 .000030588236 

 .000015282259 

 .000007637196 

 .000003817292 

 .000001908212 

 .000000953961 

 .000000476932 



.000000119219 



.OOOOOO0596O8 



.000000029803 



.000000014901 



.000000007450 



.000000003725 



.000000001863 



.000000000931 



.000000000465 



.000000000233 



.000000000116 



.000000000058 



.000000000029 



.000000000015 



.000000000007 



.000000000004 



.000000000002 



.000000000001 



23 

 24 

 25 

 26 

 27 

 28 



'•29 

 30 

 31 

 32 

 33 

 34 

 35 

 36 

 37 

 38 

 39 

 40 



Sum 



g = 

 A = 



k = 



m = 

 n = 



P = 

 1 = 



w 1= 



y = 



a' =: 



b' = 

 d- = 



e' ^ 

 /■'= 



?' = 



V = 



k' = 

 /' = 



n = 

 0' = 



.177532966576 



.098457322630 



.052967170503 



.027880229587 



.014448908703 



.0074061 80072 



.003766998147 



.001905702459 



.000960492403 



.000482856502 



.000442314856 



.000121457237 



.000060829654 



.000030448787 



.000015235790 



.000007621 708 



.000003812130 



.000001906491 



.000000953389 



.000000476742 



.000000238386 



.000000119199 



.000000059602 



.000000029801 



.000000014901 



.00000000745027 



.000000003725 28 



2 

 3 

 4 

 5 

 6 

 7 

 8 



9 

 10 



II 



12 



14 



16 



17: 



18 



19 



21 



22, 

 23 

 24 

 25 

 26 



Sum 



.0000 10001863 

 .000000000931 

 .(•00000006465 

 .000000000233 

 .000000000116 

 .000000000058 

 .000000000029 

 .000000000015 

 .000000000007 

 .000000000004 

 .000000000002 

 .000000000001 



a" =.411233516712 

 b" = .150257112895 3 

 c" = .067645202107 4 

 d" = .032403992347 5 

 e" = .015895985344 6 

 f" = .007877728730 7 

 a" = .003922177173 

 h" = .001957047643 

 i" =.00097753376510 

 k" =.000488522553 11 

 1." = .000244200705 12 

 m" = .000122085292 13 

 n" = .000061038895 14 

 o" = .000030518512 15 

 p" =.00001525902416 

 q" =.00000762945217 

 a" =.000003814712 18 

 s" = .000001907352 19 

 r" = .000000953675 20 

 v" = .000000476837 21 

 w" = .ooouoij2384-19 22 

 x" =.00000011920923 

 y" = .000000059605 24 

 z" =.000000029802 25 

 a'" = .000000014901 26 

 b'" = .00000000/450 27 

 c'" = .000000003725 28 

 291 1/" = .000000001863 29 



30 e'" zz .0000000110931 30 



31 f'" = .0000110000465 31 

 32' g'" = .000000000233 32 

 33' h'" = .0000000001 16 33 

 34,' i'" = .000000000058 34 

 35J k'" =: .000000000029 35 



36 -l'" — .000000000015 36 



37 i m'" = .000000000007 31 



38 n'" :=. 000000000004 38 



39 o'" — .000000000002 39 



40 P'" — .000000000001 40 



Sum 



f" 



.233700550136 

 .051799790264 

 .01 467 8' 13 1604 

 .004523762760 

 .001447076640 

 .1)00471548657 

 .000155179025 

 .000051345183 

 .000017041362 

 .000005666051 

 .000001885848 

 .000000628055 

 .();)0000209240 

 .000000069724 

 .00( (000023234 

 .000000007744 

 .000000002581 

 .000000000864 

 .(K)0000000286{20 



; .00000000009521 

 .000000000032 22 



; .000000000011 23 

 y" = .000000000004 24 

 .000000000001 25 



r = 



r = 

 I" = 



m' =. 

 u" ^ 



P = 

 ?" = 



1." = 



Id" 



x" 



2 

 3 

 4 

 5 

 6 

 7 

 8 

 9 

 10 



11 



12 

 13 

 14 

 15 

 16 

 17 

 18 

 19 



Prop. 1. — To Jind the sum of the sums of the reciprocal squares, cubes, &c. 

 ^c. ad irifinitum. 



By division r = - + — + - + &c, ad inf. ; hence if we make each 



-' (x — 1 ) X X x^ ' .r ' X* 



of these terms the general term of a series, and write 2, 3, 4, &c. ad inf. for x, 



we have -i- + -i- 4- \ + &c. = (table 1)a + b + c + d+ &c.; but 

 — - + 77-5 + :r~; + ^^- '"^ i"f- = 1 ' hence A + B-fc4-i> + &c. ad inf. = i . 



