338 Col. Beaufoi^ outlie [May, 



For sin. ^ h x cos. o = 4- sin. {a + ^ h) — 4- sin. {a — 4. h) 

 And sin. ± ji b x eos. (a + -i- 6 + 4- « 6) = 4. sin. (« + 4 /> 

 + n 6) — 4. sin. (cf + 4. 6) 



Therefore cos. a x sin. J- /:» + cos. (a + ^^-^ b \ sin. ^ h b 



= 4- sin. (a + 4- /^ + « b) — 4. sin. (« — j. 6) 

 But sin. (4. 6 + J- « A) COS. (a + ^ub) = x. sin. (a + 4. Z* + 



n b) — -} sin. (a — ,V />). 

 Hence the equation marked (11.) is evident. 

 Corul/arij. — Cos. a + cos. {a -f b) + cos. {a + 2 b) + cos. 



(« + 3 Z() &c. ad iniinituni = — ■ ' " , ", ' . 

 ^ ' a. sill. ^ A 



For in this case // b is infinitely great ; therefore a -\- ± b + 

 « b is infinitely great ; but the sine of an infinitely great arc niai/ 



be zero ; therefore the sum of the infinite sines = " '' """ "'"" ^ 



sin. (a — ^ 5) 

 asm. h b'' 



sin. 5 4 



Article VII. 



Oh the retrograde Variation of tlie Comjmss^ 

 By Col. Beaufoy, F.ll.S. 



(To Dr. ThoQison.) 



MY BEAR SIR, Busliey Heath, Slnnmore, April 5, 1820. 



Having completed three years' observations on the diurnal 

 variation of the magnetic needle, I have, for the purpose of 

 readily comparing the results, collected in a table the mean 

 monthly variation ; referring such of your readers as Avish i'ov 

 nK)re specific information to those numbers, of the Aiiuals of 

 P/iiloaup/iij which contain the daily observations, accompanied 

 by a meteorological table. 



By examining this general table, it appears that during the 

 first nine months of the second year's observation, the variation 

 increased ; fluctuated during the month of .January,. 1819 ;. 

 decreased in February, and again fluctuated in March of the 

 same year, since which period the corresponding mean monthly 

 variation has continually decreased, which circumstance induces 

 me to conclude that the maximum of the western declination 

 occurred about the month of March, 1819. By turning to the 

 first volume of the Annals of Phi/osop/ij/, it will be found, that in 

 the year 1657, the compass had lao variation, or pointed to the 

 true North ; consequently the lapse of timebetween.the magnetic 

 needle being at zero, and its arrival at the greatest western 

 boundary, is 162 years. In the same number of the A/uiah, it is 

 also recorded, that the declination of the needle in 1.080 was 



