&c. &c. 



x((V,)*-(c 1 ) s )<(V 2 ) 2 -(o s )')-((V p ) 2 -(< ;p r), 

 where, in general, any system of values 



C V C< 2> C 3> ••• C p-V C p 



represents 



1 1 1 



^i /*p-2 /*/o-i 



Then the largest root of X=0 is a superior limit, and the 

 smallest root of X = is an inferior limit to the real roots of 

 y*#=0; and if X=0 has no real roots, neither will^a^O have 

 any. For the complete demonstration and some further deve- 

 lopments of this theorem see the forthcoming number of Ter- 

 quern's Nouvelles Annates for the present month. 



Cafe Militaire, Clermont Ferrand, 

 July 15, 1853. 



XX. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 6.9.] 

 April 28, A PAPER was read, entitled " On the Application of 

 1853. -£*- the Law of the Conservation of Energy to the De- 

 termination of the Magnetic Meridian on board Ship, when out of 

 reach or out of sight of Land." By W. J. Macquorn Rankine. 



The author states that, assuming that when a ship is swung com- 

 pletely round, so that her head bears exactly as it did at first, the 

 magnetism of the ship, and that of the compass-needle return to 

 their original condition, the following theorem is necessarily true : — 



The mechanical power developed by the mutual action of the ship and 

 of the compass -needle during a complete revolution of the ship, is equal 

 to zero. 



If £' be the apparent magnetic azimuth of the ship's head, east of 

 north ; a' the corresponding apparent magnetic azimuth of a distant 

 fixed terrestrial object (or where no such object is visible, of a star, 

 corrected by calculation for its apparent diurnal motion) ; a the 

 true magnetic azimuth of the same object, so that a— a! is the 

 westerly deviation of the compass-needle ; then the above theorem 

 is expressed symbolically thus : — 



0=1 sin(a— a').d? f =sinal cos a' .aX— cos at sina'.c??'; 

 Jo Jo Jo 



