P R K 



the population had died annually ; no rector \va> kopt of the birth? ; 

 What was the annual proportion of them during that period *i 

 Here I = '.',\.\. 



The annual proportion of births was about th. 



Supposed Population of the World. (Enc. Brit. Suppl.) 



ElirODe ^r^ff t^^r^r^f^rf^ff^f^f^^r^rr^f 185 000 000 



Asia (with Australia and Polynesia) ~~, 270,000,000 



Africa ~~ 55,000,000 



America v~~~~~~~,~~~~~~ ,~~ ,.~~ : ~ 40,000,000 



350,000,000 



POWERS of numbers. See Involution. 

 POWERS Mechanical See Mechanical Powers. 

 PRECESSION of the Equinoxes. (Woodliouse* Vince, Play fair.) 



I. The mean annual precession = 50",34, which gives nearly 1 for 

 the precession in 71| years, or about 25745 years for the entire revolu- 

 tion of the pole of the Equator round that of the Ecliptic. The part of 

 the precession arising from the action of the sun 15",3, that from the 

 moon = 35". If the effect of the sun bo reduced to 12",5 ; that of the 

 moon will be triple of it, whicli is agreeable to the latest results deduced 

 from the theory of the tides. 



The precession affects the situation of stars in Declinatiou or North 

 Polar distance, and in right ascension ; hence the following Formulae. 



Annual Precession in Declination. 

 This = 50",34 X sin. obliquity X cos. star's R. A. 

 Cor. When the right ascension (R. A.) is between 90* and 270, the 

 declination is diminished by the effect of precession. And when the R. A. 

 is between and 90, or between 270 and 360, the declination is 

 increased. 



Annual Precession in R. A. 



This = 50",34 X (cos. I + sin. I X sin. star's R. A. X tan. star's decli- 

 nation) where I obliquity of ecliptic. In this expression the first part, 

 50" ,34 cos. I is common to all stars. 



Cor. The precession in R. A. is nothing when the angle of position is 

 aright^; it is also positive when that / is acute, ai^d negative when 

 obtuse. 



II. Precession Solar t inequality of. 



The mean annual precession has been stated at 50",34 ; but it cannot 

 have been equably produced, For the sun is sometimes in the Equator, 

 210 



