385 



the last term in the table : it will therefore resolve into square num- 

 bers any odd number up to 9121 + 190=9211. 



With reference to the mode in which the intervals in the table 

 maybe filled up, the author states the following general theorems re- 

 lating to the sums of three square numbers, by means of which the 

 roots may be varied, and yet the sum of the squares remain the 

 same. 



Theorem D. — If any three terms of an arithmetical series, and 

 omitting the 4th term, the three following terms be arranged thus, 



a + b, a + 2b, a+6b, 

 a , a + 4b, a + 5b, 



the sum of the squares of each set of terms will be the same. 



Theorem E. — If four numbers in arithmetical progression be placed 



thus, a , a-\-2b, 



a + 4b, a + 6b, 



and the sum of the 1st and 4th be divided into two parts whose dif- 

 ference shall be four times the arithmetic ratio, as a-{-7b — (a — b), 

 and the parts be placed with the terms, the greater with the less, 

 and the less with the greater, thus, 



a , a + 2b, a + 7b, 

 a—b, a-\-4b, a-\-6b, 



the sum of the squares will be equal. 



Theorem F. — Let two numbers which differ by 2n be placed thus : 



a-\-n, a-{-n, 



a — n, a — n, 



then if the sum of the four (4a) be divided so as to have the same 

 difference (2w), and the parts be placed, the less with the greater, 

 and the greater with the less, thus, 



a-\-n, a-\-n, 2a — n, 



a—n, a—n, 2a-{-n, 



the sum of the squares shall be the same. 



The author illustrates this part of the subject by deducing six 

 forms of roots whose squares =197. 



January 12, 1854. 



The LORD CHIEF BARON, V.P., in the Chair. 



Commander Kay, R.N., was admitted into the Society. 



A paper was read, entitled " On some New and Simple Methods 

 of detecting Manganese in Natural and Artificial Compounds, and 



