200 Wisconsin Academy of Sciences, Arts and Letters. 



employed only for the signal-nights, or other times when the last degree 

 of precision is required. 



8. In making the least square reductions the weights at different declina- 

 tions should be combined with the coefficients for azimuth and coUination; 

 and the imknowns should be in the form of corrections to the values derived 

 from the preliminary reduction. 



Jacobi's Theorem. 



Note. — [I give this in the original Latin, as it is very important, and 

 seems not to be well known to mathematicians.] 

 Proponantur aequationes: 



ax-|-a'x,-[-a"x2 -fa*- -^x^ =1 



ajx+a'-(^x,-f a"j x^ +a,^^^ x^ =1^ 



^p^+S^^i+S^a +^^''^^n=lp 



Quarum numerus incognitarum numerum excedat; e quolibet systemate 

 n+1 aequationum. praecedentium valor incognitae eruatur atque per quad- 

 ratum Determinantis eius systematis, RR, multiplicetur; quibus factis pro 

 singulis aequationum jDropositarum combinationis omnium illorum produc- 

 torum summa jDer summam omnium RR dividatur: eruitur incognitae 

 valor idem atque invenitur, si aequationes propositae per Methodum Min- 

 imorum Quadratorum tractantur." 



Observandum est, valores omnium incognitorum qui ex eadem aequa- 

 tionum propositarum combinatione proveniant secundum Prop, praec. per 

 eandem quantitatem RR multiplicare, quam ideo in applicationibus ad 

 Methodum Minimorum Quadratorum convenit, appellare Pondus Com- 

 binationis, a pondore valoris incognitae bene distinguendum. 



The theorem is thus given in Crelle, vol. 22, p. 316; it is also given inde-. 

 pendently by Mr. J. W. L. Glaisher in Vol. 40, of the Monthly Notices, p. 

 607. 



