13 



REPORT — 1873. 



four, and for 1843-5 with seventy-four numerically different equations of 

 the form 



n = a . cc^ + h . (x,, + c . a,^ + d. a,^ + e . ec.+f . a^-\-(j . a,.-\-h . a^, 



whicli directly to satisfy was of course in both cases impossible. But in 

 order to determine those two sets of the eight unknown aj, a, . . . . a^, wliich 

 according to the rules of iirobabilitj had to be assumed for the first and for the 

 second of the said years, it was necessary to supply the just mentioned 

 theoretical form of the conditional equations by the practically possible 

 assumption of 



«=(— n+« . ct^ + h . a.., + c. . oi^-\-d . a^ + e . cc.-[-f . a.^-\-fj . a-,+h . aj . Vj), 



p and V in this expression being meant to stand for the so-called iveic/ht of 

 every value of n, and for the error to be supposed in it. 



If, then, [ ] indicate generally a sum of algebraically similar terms, and 

 if the assumed values of the error v be regarded as functions of the un- 

 known, we shall obtain the most probable values of a^, a, . . . . a^ by the 

 solution of the following eight final equations under (G), which in their turn 

 are but evident consequences of the general principle under (©) 



(O) [v^] = minimum. 



w . -^ =o——[anp'] + lacqj']a^ + [abp']a^ + lac2)']a^ + ladp']a^ 



+ [««P]«5 + C«fp]«6 + lm'']»7 + [('h'l^s' 



-1- [ bep-ja, + [ hfp-]a^ + \]>cjp-]a, + [67ip]o,. 



\v • ^1 =0= - [ cnp-] -f [ capy^ -\- [ chpyi.^ + [ ccp']a^ + [cdp ]a, 

 + [ cep']a, -h [ cfp-]u, + [ qip-]a, + [ cl,p-]a^. 



\v.~\ =o=—ldnp\ + \d(ip]a^^ldhpyi.^-\-[dcp~\a^-\-[ddp']a^ 



.1 = = - [ enp-] + [ ra/>]ai + [ cl>p']<i., -|- [ ecp']a^ + [ edp^t^ 



+ [ <^fi^>5 + [ C^i^Jac + [ m^l^i + [ clipl^^i- 



1 =o=-ifnp'\ -\- U('F\«-i + [/^P]"2 + [/'P]«3 + L/'^]"* 



+ [hep]^, + [7*^>a + U^gp^^-, + \]dip'\a^. 



"We have here retained the general form of this prescription for calculating 

 Oj, Oj. . . .Og, though when employed for the year 1811 it became simplified 



(6) 



< 





dv' 

 da, 



dv_ 

 dtt.^ 



dv 

 da. 



