INTERPOLATION. 



SuA an example is the mofl fimple that can be conceived loncritudc in feconds and tliirds, hay-ng firft found 7/7j5«';w?- 



in interpolation, but fimple as it is, it is all that is required in trlcally the fame for every degree. In general, whatever he 



aRronomy, even for the motion of the moon, which is the the nature of the calculation, it will be fufficient to afcert-iii 



moft irregular of all the planetarv bodies; fuch at leaft is rigoroufly by trigonometry, or xjthervrife, terms at fuch a 



the affertion of Lalandc in the ariicle of the Encyclopedic diilance that their third difterences may become zero, or fo 



above quoted. fmall that no fenfible error will arife in confiderkig them as 



Nou' knowing the above numbers, or obferved longitudes, ["^h and then all the i«termeaiate terms may be luophed 

 for ever^- .2 hours, it will be eafv, bv means of the above ^y^^f ^^'''°P' ^}-}^^^ f'^' ^^^" >"ve(bgated Lal^nde h,-.s pub- 

 rule, to 'find the fame for every f.x hours ; for fmce, in this l>«^ed m the " Connoiffance des Terns for t„, a very 

 cafe, the fecond differences onght to be only one-fourth of commodious table for abrmgmgthefe kmd of operations and 

 thofe above, that is 36, it will therefore only be neceiTarv to 'here .s another ft.ll more extended m the Recue.1 des 

 conftruft a feries of numbers beginning at o, whofe fecond ^■''J°f^°'^ ^^'^'',""- ^ ^ , ,... .^ . 

 ,(j- /: „A \ r E.^^ A t^^^ /J,-,ll Ko -,<;• The fame theorv of fecond dilterences may alio be con^e- 

 omerences are jo, and whole iecond term (nail be 70 ; . ,■ j ' i. 1 1 • 1 tr \. 



^ . T^tnnfN- :5TM-. ifiH trt mryt^f\ /^^Ipii atTrtnc anf nh(pffuatmnQ_ that- 



„.^ntly applied to correcl calculations and obfervations, that 



therefore ZzJZJ^ = 21 is the firft term of the firft dif- is, by afcertaining the uniformity of the remainders or diF- 



2 ferences. 



ferences ; and all the other terms will be found by adding For example, a feries of obfervatJons being fubmitted 



■fucceffively to this, the conllant fecond difference 36 ; thus to this tell, ought to have a certain order of their 



n. ijjj- 21 rn o5 iiQ i6' 20t differences Uniform, and if in any place fucli a uniformity is 



Nos. orlong. 0.2W 78', 171, 300, 46}!- 666. not obferved, one difference being greater, or lefs, than 'aa- 



° ' •> ^ -> other, It may be concluded with certainty, that fome error 



But if, inft.-ad of interpolating one term between each of jj^^ ^^^^^ committed in the correfponding obfervation ; the 



the given nuntbcrs, it had been required to interpolate two correction of which may be readiiy made, without repeating 



terms, then we muft have taken a ninth part ot t!)e above ^j^^ obfervation ; Lalande has alfo given, in his memoirs 



feaond difference, i>/z. 16, and found from heiTce a feries above-mentioned, general formula of corre£tion, for all 



of numbers whofe fecond difference Ihould be 16, and the j-.,^,, ^^f^^^ g„^ ^.j^j^h ^^y ^^ app,;^j .^j^j^ ^j^^ greateft fa- 



third term of the feries = 78. Now in order to hnd this ^-^^ ^^^j ^j^^ corrldion computed to the utmolt poffible 



feries,' it will be fufficient to hnd the firft term of the lit dil- accuracy. 



ferences, as all the reft that is required may be determined Qu nearly the fame principles as the foregoing, the ex- 



from this ; fuppofe, then, this term to be x, which is alfo the traaion of roots, as the fquare root, cube root, &c. may 



firft term of the required fenes, then the iecond will be ^^ ^fr^^^^ ^jj,, ^ ^^f^^ ^y knowing certain equidiibnt 



2.V + 16, and the turd 3. r + 48; whence 3 .r + 48 roots. The application in this'cafe being extremdy fimple, it 



= 78, or.f::= 10; which gives for n,ay not be amifs to enter a httle into the explanation 



the iftdiff. 10, 26, 42, 58, 74, 90, ic6, &c. of it. 



Nos. orlong. o.io, 26, 78, 156, 210, 300, 406, &c. _ , 



r -1 ■ ■ 1 1 ■ I ^ ■ u Let then 



and on fimilar principles may the interpolation m any other 



cafe be tffefted with the fame degree of eafe and accuracy ; r% 1 jv _ ^ j L _ _i I- — — _ ^° 4- <tc 



that is, generally, in order to interpolate any number of ^' ' j .r' 9.1' Si .v' 243 -v' 



terms n, between any two given terms of a known feries, /; y _ ^, 

 we mull divide the fecond difference of the given feries 



ty (n + I ) ', in order to have the fecond difference of the / ^5 j u _ _^. I I > ^° _ g.^^ 



new feries; and then again in order to obtain the i ft dif- 3 .v" 9.*' Sl.v' 243 .v" 



ferences, we have this formuk ; where ./ = i ft term ill diff. reprefent the cube roots of any three cbnfecutive numbers, 



of the given fenes; and rf" = 2d difference new ierics, tlie differences of which are 



Iff term .ft diff. =. -JL, ± 'if ; T J_ ^ J_ ^ _1_.^ ^ ^o__ _^ ^^_ 



■which may be more conveniently exprefl'ed in terms of the j i i 5 10 



tft and 2d differences of the given feries as follows; d z= ift 1-7"? [73 "^ Rj TP^ ^2 v" ^ ' 



<ljfference, {!'■= id difference of tlie given feries, then the 



ift term ift diff. of the new ilrics, becomes 2d diff. -^ -j- ^° .■ &c. 



. _ 2d' {n -r I) + K d' *->•*'■ -43-'''' 



— 5 (11 + I V ' I" the fame way, was it neceffary, we might have found 



' the third and fourth differences, and fo on ; but in the.cale 



and this term being thus determined, all the others are ,ve (liall fuppofe, of x' > 1000, or .r -- 10, this fecond 



readily obtained ; and confequently alio all the mtermediate difference is fufficiently exnft, and even the fecond term of 



terms which were to be interpolated. The confideration of this fecond difference is fo fmall, that it may be omitted 



fecond difterences, which render the interpolation fo extremely without affeding even the eleventli place of decimals, for 

 eafy, i-', as wc before obferved, iufhciently-exaft for the 



greater part of aftronomical calculations, particularly in the taking .*■ = 10, we have — tS— = ^-— .which 



conftriRHion of tables ; it was tlius that Siiarp in 169J cal- 243 x • 2430000000000 ' 



culated his table of riglit afccnfion and declination for every ,,.],gn converted into a decimal, will not give an effeftive figure 



degree of latitude and longitude ; having iirft calciilated tn- ^^{^^^ ^^^ j jt,, place ; we may, therefore, without any fenli. 



^onometncaUy the fame for every fiftli degree, the other inter- ' •' 



mediate degrees haying been afcerlained by means of tlie ble error, call — - the fecond difference, which will remain 



theory of interpolation. Mouton alfo, on fimilar principles, 9 •'■' 



calculated the declinntion of the fiKU for eveiy inini>te of cenflant for feveral terms, while our extradion is not carri-d 



farther 



