INVESTIGATION OF CERTAIN THEOREMS, &C. l%y 



II. 



Jnvejligation of certain Theorems relating to the Figure of the 

 Earth. Bjj John Playfajr, F. R, S. Edin. and Prof ejjbr 

 of Mathematics in the Univerjltj/ of Edinburgh. 



(Concluded from Page 11 6 J 



21. A REMARK, that is in no danger of he'mg reckoned This methoJ by 

 h_ypothetical, is, that the conclufion derived from the compa- j^g^^pg^^kular 

 rifon of degrees of the meridian, with degrees of the circle is more errone- 

 perpendicular to it, becomes of neceffity more liable to error ^^^ []^^j^^^|^^' 

 as we advance into higher latitudes. The reafon is, that 

 whatever error is committed in determining the magnitude of 

 D^— D, muft be multiplied into' the fquare of the fecant of 

 the latitude, in order to give its full effe6l in changing the 



value of the fraflion ^. For it has been fhewn that ~ = 

 a a 



— [ — --7~ J fee ^(pi now, if we fuppofe the error committed 



in afcertaining D'— D to be in all cafes the fa;ne, the error 



of the fraction — -—^ will alfo be in all cafes nearly the fame, 



the denominator D' being but little affeded either by the fup- 

 pofed error, or by the change of latitude. But this'^error, 

 which may thus be confidered as a conftant quantity, when 



1 . . c 



multiplied into — fee *<?>, gives the variation or error in -, 



which error therefore increafes, co'teris paribus, a§ the fquare 

 of the fecant of the latitude, fo that, on approaching the pole, 

 it increafes without limit, and is ultimately infinite. Compa- 

 rifons of this kind may therefore be expelled to give refults the 

 more accurate the nearer they are to the equator, under which 

 circle they will be the mod accurate of all. Here, again, 

 however, another circumftance muft be taken into- confidera-* 

 tion, viz, that the method of afcertaining the differences of 

 longitude by the convergency of the meridians, fo convenient 

 in furveys of this kind, is applicable only in high latitudes. In 

 a trigonometrical furvey, therefore, of a country lying much 

 farther fouth than Britain, a different method of afcertaining 

 the longitudes of places muiJt neceflarily be adopted. 



22. The 



