158 



SCIENCE. 



[Vol. II., No. 27. 



the next instrument ; and so on tlirongh the 

 series. Instead of sending down a messenger 

 on the wire, a propeller-wlieel has also been 

 arranged to open the jaws, so that either 

 mefhod may be eniploj-ed. 



The question arises, wliether, with these ex- 

 cellent methods of using the instrument, the 

 Negretti and Zambra thermometer cannot be 

 made to record accuratel}' to tenths of a 

 degree. It would seem, that, by giving it a 

 short range and a comparatively long tube, 



this might be done. If not, the most delicate 

 observations for sub-surface temperatures will 

 pi-obably have to be made with some form of 

 apparatus, which, like that used b3- Professor 

 Ekman, brings the water to the surface in a 

 case covered with a material througli which 

 heat cannot readilj- pass, or else bj- sending 

 down a thermometer enclosed, like Capt. 

 Rung's, in a thick case of non-conducting 

 material. _R. Hitchcock. 



Loudon, June 1, 1SS3. 



nEAL ROOTS OP CUBICS. 



THEOREM I. 



[In the equation x^ + Ax" + i? = 0, when 

 the roots are real, A and B have opposite 

 signs ; and simultaneously changing the signs 

 of ^ and B changes signs of roots of equation.] 



Assume x = a, x = b, x = =-. 



, (a^+ab+b-)x-+—-j {a-¥) = 0\ 



(1) 

 (2) 



a+b ' ' - ■>- ' ^_j_^^ 

 and, changing signs of roots, 



x^+-^(a-+ab + ¥)ar ~ {a-b-) = 0. 



a+b a+b 



Since the factors (a^ + ab + b'-) and (crb-) are 



positive when the roots are real, whatever the 



sign of , A and B will have opposite signs, 



and, from (1) and (2), simultaneously chan- 

 ging signs of A and B changes signs of roots 

 of equation. 



THEOREM n. 



rA^ B n 



^ is greater than — - in quantitj*. I 



Assume -^, -r^r— I > —: -r^ (3) 



V 3(a + 6) / 4(a+6) ^ ' 



> a-b- 



a+b\- 



f a^+ab+b- y 

 "^j > ab (Algebra) 



but 



hence inequalit3' (3) is true. 



A 

 From (1). omitting the term — , 



(-f4T^ ^U 



B A' 



-4+27 



V^l + 



Jl 



i 



B A? 



"4+27 



V-1 



(4) 



and, from (2), 



= 1 + 



J^ 



+^ 



J! 



/=!■+ 



^ 4+27 ,--N 



(5) 



