310 



Royal Society. 



always lie within such limits as it would have had if determined by 

 actual observation from the same experience as the data. 



The proof of these propositions rests upon certain general theorems 

 relating to the solution of a class of simultaneous algebraic equations, 

 and, auxiliary to this, to the properties of a functional determinant. 



The following are the principal of those theorems : — 



1st. If the elements of any symmetrical determinant are all of 

 them linear homogeneous functions of certain quantities a v a 2 , . . . a T 

 — if the coefficients of these quantities in the terms on the principal 

 diagonal of the determinant are all positive — and if, lastly, the coeffi- 

 cients of any of these quantities in any row r of elements are propor- 

 tional to the corresponding coefficients of the same quantity in any 

 other row, then the determinant developed as a rational and integral 

 function of the quantities a v a 2 , . . . a r will consist wholly of positive 

 terms. 



And, as a deduction from the above, 



2ndly. If V be a rational and entire function of any quantities 

 x v x 2 , ... x ni involving, however, no powers of those quantities, and 

 all the coefficients being constant, and if in general V { represent the 

 sum of those terms in V which contain x { as a factor, and V$ the 

 sum of those terms in V which contain the product x t x j} then the 

 determinant 



V. V. 



V, . . . V. 



V 13 . . . V, 



v« . . ■ v 2 



v„ V. 



will on development consist wholly of positive terms. 



3rdly. The definitions being as above, and the function V being in 

 form complete, i. e. containing all the terms which by definition it 

 can contain, the system of simultaneous equations 



y— Pl> y— P2> 



--Pn, 



in which p v p 2 , . . . p n represent positive proper fractions, will admit of 

 one, and only one, solution in positive integral values of a? lS x 2 , . . . x n . 

 4thly. The function V being incomplete in form, u e. wanting 

 some of the terms -which it might by definition contain, the system 

 of equations 



v 



■-p, 



■p» 



f=* 



will admit of one, and only one, solution in positive integral values of 

 x v a? 2 , . . . x n , provided thatp l9 p 2 , . . .p n , beside being proper fractions, 

 satisfy certain conditions depending upon the actual form of V. 

 These conditions are expressible by linear equations or inequations 



