682 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1780. 



was at 50°; and the lowest was — 11°, or 11° below zero, viz. on Jan. ]g, on 

 which day l6 or 17 German soldiers were frozen to death in crossing the ice on 

 the Lake St. Pierre, a lake near Trois Rivieres ; and about double the number 

 were frost-bitten, many of whom lost their feet and hands. 



By or (It 



tr 



XVI. Meteorological Journal kept at the House of the Royal Society 

 of the President and Council, p. 279- 

 Of this usual journal of the thermometers, barometer, rain, winds, and wea- 

 ther, kept at Somerset-Place, for the year 1779, the synopsis is contained in the 

 following table. 



The mean variation for the year was 22° 4'-^. 

 And the mean dip of the needle 72° 21'. 



XVII. Theorems for Computing Logarithms. By the Rev. J. Hellins, p. 307. 

 After some introductory remarks on the utility of logarithms, Mr. H. adds, 

 the following theorems, inferior to none as to convergency, are useful in deducing 

 the logarithms of great fractions from those of small ones, or the logarithms of 

 small numbers from those of great ones. 



P + 1 



.2/) + 2y 



Theor. 1 . The log. of ^-^ = 2 X log. o(tL^ + log. 

 Cor. If q := 1, and we write nforp, the theorem becomes 



n + I _ „ , 2n + 2 , , (2n + i)' 



of 



C2p + 'lY- 



(2p + qV - qq 



log. '"4-- = ^ log- iirri + '°^- {fin+i)"^- i ' "^^""^^ expression perhaps is of 

 more frequent use than that above. 



Theor. 1. Log. ^-^ = 2 log. '^ + '' 



'2p 



^ (2p + qY - 



vv 



