

iiK RATING TO THE FIGURE OF THE EwVRTil. W^ 



In the fame manner that a has been found, we will obtain To iind the axes 



. mD'cof^"" ftt'afphcroid 



^ ~ -- — -^ -— — -. irom companhg 



/, ^^ r ^A /^ adeg. ofthe 



(^ 1 - -p fin (?=» j v/~ xnerfd. with one 



If we examine thefe formulas in the extreme cafes, viz. 

 when <p zz 90®, and when ^ zz 0, we fliall have in the for- 

 mer cafe a rz — , becaufe cof <p ZZ 0, and alfo D/ zi D, fo 



D 



that 1 — ^ fin 2^zz 0. Here therefore a is indefiqite, ,and 



may be of any magnitude whatever ; and it is evident that this 

 is the refult which the formula ought to give : becaufe at the 

 pole, or when ^ zz 90°, the perpendicular to the meridian 

 is itfelfa meridian, and therefore the meafurement ofthe two 

 degrees, D and D^ is but iTie fame" with, the meafurement of 

 one degree. 



When (p zz Oi that is at the equator, the circle perpendi- 

 cular to the meridian is the equator itfelf, and we liave theii 

 a ZZ mD\ a being determined in this cafe by the degree of the 



a W ' ' 



equator alone^ Here alfo we have --*- zi V'pT' which 'i^ 



known to be true. 



18. The preceding forrnulal m^' be rehdeVed more {iraple> 

 if we aim only at an approximation, which ihde^d is all that is 

 neceflary in this inquiry. Since c denotes the compreffion, or 

 fmce a-^czzh, and therefore a^ — 2aCZZ b^ nearly, confe- 

 quently the radius of curvature of the meridian at F, that is 

 j^ _ g^ (q^ — 2ac) a^ (a—Qc) 



(a^ — 2ac fin <|)*)* aM I — -^ fin (p^)^ 

 ^ a 



Sc 

 (a — 2(^) (1 — ^— ' fin <?*), or mD s= a — 2c -f. 3c {m (p». In 



the fame manner mD'= a -|- c fin (?>*. From thefe equations 

 we obtain, rejecting always the higher powers of c, 



C =. — ^ i, a = mD' i— —^ — —_ ; and 



2cof(2)* 2 cof 9^ 



c 



D' — D 



« 2D' cof <P^ 



Thefe formulas may be transformed into others a little mor« 

 convenient for computation, by putting fee <?* infiead of 



Vol. VII.— February, ISOt, I 1 



