1862.] 209 



13 to 101, and beginning with 13, a number be taken from each line 

 alternately, a series of the first kind will be discovered, viz. one in- 

 creasing by 2, 4, 6, 8, 10, &c., and every term in that series will 

 become the first term of another similar series ; so that every odd 

 number in the small squares (except those in the top row) will be 

 terms in 2 such series, and the indices will be as marked in the 

 right-hand upper corner of each square ; the numbers in the margin 



1, 3, 5, 7, &c., and 0, 2, 4, 5, &c., are the indices of all the series, 

 parallel to them. 



Now 51 = 0, 1,5,5. No two roots differ by 9 ; but 5=5 ; and 

 5 + 5 1 = 9. As 5 = 5, all the numbers below 51, that is 53, 59, 

 &c., are divided into 4 square numbers, whose roots appear in the 



diagram. Again, 35=4 2 , 4 2 + (?)=3 2 , 5 2 + 1, and 35= 2 1,0,3,5, 



which gives 7, the index of 35, as a term in the series increasing by 



2, 4, 6, &c. ; and therefore every term may be resolved into 4 

 squares ; but as one series crosses a set of series, it at length fur- 

 nishes the index, thus, 51, 37, 31, on arriving at 



-0, -1,5, 5 -1, -2,4,4 -2, -3, 3, 3 



31. 2+3=5, the index of 31 as a term in the series increasing by 

 4, 8, 12, &c. ; and 19 therefore equals 0, 1, 3, 3; and 1, 3=4 

 (the index of 19 as a term in the 3rd series) ; 1 1 =0, 3, 1, 1. 



The method by which the division of certain numbers into 4 

 squares is here accomplished applies to all numbers of the form 

 2n + 1 ; but as the first term increases, the methods also multiply, so 

 as to afford increasing means of division, which must be the subject 

 of a future communication. 



XXVI. "On the Oxidation and Disoxidation effected by the 

 Alkaline Peroxides." By B. C. BRODIE, Esq., F.R.S., Pro- 

 fessor of Chemistry in the University of Oxford. Received 

 June 19, 1862. 



(Abstract.) 



A preliminary notice containing an abstract of the greater portion 

 of this paper has already appeared* . 



Having shown that the alkaline peroxides are capable of acting 

 either as agents of oxidation or reduction, the author proceeds to 

 * See ' Proceedings,* vol. xi. p. 442. 



