288 PROCEEDINGS OF THE AMERICAN ACADEMY 



Five Hundred and Thirty-fourth Meeting. 



May 10, 1864. — Monthly Meeting. 



The President in the chair. 



The President called the attention of the Academy to the 

 recent decease of Dr. John Ware of the Resident Fellows, 

 and of General Joseph G. Totten of the Associate Fellows. 



Mr. Oliver presented a partial investigation on the best ap- 

 proximate representation of all the mutual ratios of k quan- 

 tities by those of simple integers. 



When k = 2, it is well known that the method of continued frac- 

 tions gives every pair of integers whose ratio is more accurate than 

 that of any simpler pair : [as well as every proper approximate ratio, 



i. e. every approximation — ^ to — m which a certain further crite- 



rion that involves either the previous or the next proper approxima- 

 tion, as that ?«/'■* 7nS''^^'> — m./^ m/'"'-^> = ± 1, is satisfied.] 



But when k > 2, the | k {k — 1) separate ratios of the given 'quan- 

 tities (mi , . . nil) can seldom be very accurately represented by those 

 of the same small integers m\ , . . m\. ; and a different kind of crite- 

 rion is needed to show how far we must rectify either ratio at the 

 expense of others. This criterion of excellence depends, of course, 

 on the smallness of some such function F of the ^ k (k — 1) deter- 

 minants tn^ in, — m, m\* and the 2 k general magnitudes (mj, . . m\), 

 as shall be independent of the interchange of (m,. , m\) with (m^ , m\), 

 and shall vanish when all the determinants do, and not otherwise 

 when (mj , . . w^\.) are real and positive. But it remains to fix the 

 precise form of V by other considerations ; for instance, by the neces- 

 sities of some particular problem. 



Here the simplest case of the problem " to determine the best 

 approximate weights of a set of linear equations," seems hkely to 

 give a very natural criterion, whether others equally natural exist 

 or not. Let (mi , . . m^) be the weights of equations (x = Xi, . . 

 X = x^), from which to determine x. 



Let S=mi-\- ..mk, *S" = m\ -\- .. m\. 



* Usually of the squared determinants, so as to be independent of the determi- 

 nants' signs. 



