Intelligence and Miscellaneous Articles. ■ 149 



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 he arranged thus, 



a-\-h, a + 2b, a + 6b, 

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



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



TkeoremE. — 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 + 76 — (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 2m be placed thus : 



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



a — n, a — n, 



then if the sum of the four ('2a) be divided so as to have the same 

 difference (2n), 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. 



XXII. Intelligence and Miscellaneous Articles. 



ON THE OCCURRENCE OF NICKEL AND COBALT IN SOME MINERAL 

 SPRINGS, AND ON A METHOD FOR THEIR ISOLATION. BY 

 OSSIAN HENRY. 



MAZADE some time since stated that he had found in the 

 chalybeate springs of Neyrac and its ochreous deposits, 

 titanium, glucina, cobalt and nickel. In consequence of this state- 

 ment, the author has tested several chalybeate waters for nickel 

 and cobalt, and ascertained the presence of these two metals by 

 the following process : — 



To a large quantity of the water a slight excess of carbonate of 

 soda is added ; the fluid is then allowed to stand in the air until the 



