§6 Experiments with Potassium and Sodium. 



a''§+§7'=a2-j-g2y2_gj^o,g2y2 and as many corresponding values 

 for 2b. So for any greater number of prime factors, a2gay2^2 ^^^ 

 Thus we see that an hypothenuse or a base may be found, upon 

 which may be constructed any required number of right angled tri- 

 angles, whose sides shall all be rational integers. 



5. When P is neither a square nor the sum of two squares; the pro- 

 duct P*Q cannot contain the sum of two squares, except where Q 

 takes theforms P(m='+w='), F = {m^ ■i-n''), 4P{m^ +n^), &tc.— that 

 is, when P and Q have a common divisor. 



For if Q be separated into any two parts whatever, (when P and 

 Q are prime to each other,) it is manifest that P;r+Pt/=P'Q is not 

 the sum of two squares, whether x and y are squares or not. 



6. Every number which is the sum of two squares that are prime 

 to each other, is either a prime number, or divisible only by factors 

 which are also the sum of two squares. 



For, if we suppose that — p — =Q ;• then a^ -\-b^ =P-Q,, which 



has been shown to be impossible. 



Corollary. Hence, if A^ =D^ -{-C'^ ; then A = c^-{-d^, as was 

 asserted in a previous remark. 



Art. XL — Experiments with Potassium and Sodium; by Julius 

 T. DucATEL, M. D., Professor of Chemistry, in the Medical De- 

 partment of the University. of Maryland. 



Seruli.as says that a piece of Potassium put upon a bath of Mer- 

 cury, gradually amalgamates, acquiring a rotary motion, due to its 

 action on the water in the atmosphere, which evolves hydrogen. In 

 dry air the amalgamation takes place without motion. But if pieces 

 of Sodium be thrown upon Mercury, they are again thrown off with 

 slight explosions, accompanied with light and caloric. On the other 

 hand. Potassium burns on the surface of water, while Sodium decora- 

 poses it without producing combustion.* These effects of the two 



* I observed many years ago, that on hot Water sodium scintillates, and even bla- 

 zes. — Ed. 



