188 Mansfield Merrhnan — List of Writings relating 



A reply to Reuschle's ci-iticisn)S. Encke tries to explain that in 

 the two expressions 



2l q) {x) z= 1 and /J," </> (a*) dx = 1 



the symbol 1 has difterent meanings. He also shows that Reuschlb 

 had tailed to understand his i^roof of the arithmetical mean. 



1844 DoNKiisr. An Essay on the Theory of the Combination of 

 Observations. Trans. Ashmolean iSoc. (Oxford. 8vo). — French 

 abridgment, ' Sur la theorie — .' in Liouville^s Jour. Math. 1855, 

 Vol. XV, pp. 297-322. 



DoNKix attempts to establish " une espece de Statiqne mc'ta- 

 physiqiie snr des preuves de la meme force que celles qu'on emploie 

 en deduisant, a priori, les lois de la Statique ordinare." The word 

 " force" is taken to mean " tout motif qui nous porte a alterer la 

 valeur attribuee a une quantite," and to these " forces" the princij)les 

 of centre of gravity of bodies, of virtual velocities, etc., are applied, 

 and the usual rules for the adjustment of observations by means of 

 normal equations, weights, mean errors, etc., are deduced. No law 

 of facility of error enters iuto the discussion. 



Doinkin's reasoning does not always seem to me clear or rigorous. 



1844 Ellis. ' On the Method of Least Squares.' 7)-ans. Camb. 

 Phil Soc, Vol. Vin, pp. 204-219. —Also Bllis's 3Iathematical and 

 other Writings (Cambridge, 1863, 8vo), pp. 12-37. 



In this pa])er it is attempted "to bring the different modes in 

 which the subject has been presented into juxtaposition, as that the 

 relations which they bear to one another may be clearly ap])re- 

 hendcd." 



Ellis first takes up Gauss's proof of 1809. He considers that 

 Gauss is not justified in assuming that the rule of the arithmetical 

 mean gives the most probable values, and he shows that besides mere 

 convenience no satisfactory reason can be assigned why it should be 

 so regarded. His remarks on this point are extremely valuable and 

 soumi See 1872 Glaisher. 



Laplace's demonstration is taken up and presented in a different 

 but greatly simplified form, extended to the case of any number of 

 unknown elements. Gauss's second proof of 1823 is also analyzed 

 and the conclusion arrived at that "nothing can be simpler or more 

 satisfactory." Lastly Ivory's three proofs (1825-6) are discussed 

 and their illogical character clearly exposed. The paper is one of 

 the most valuable in the theoretical literature of the subject. 



1844 Jacobi. 'LTebereine neue Auflosungsart der bei der Methode 

 der kleinsten Quadrate vorkommenden lineiiren Gleichungen.' Astron. 

 Nadir., Vol. XXII, col. 297-306. 



An abridged method for the solution of certain forms of noruuil 

 equations. 



