232 Reduction of Latitude. 



tangent is a maximum in the present case, therefore, to find the lat- 

 itude of the place where the reduction of latitude is the greatest, we 

 have only to put the first differential coefficient of tang. (5 equal to 

 zero, and the resulting value for tang. 4^, will give the latitude 

 sought. Thus, formula (6) being differentiated, and the first differ- 

 ential coefficient being put equal to zero, we obtain 1 — (1 — ay X 



1 rt. 



tang.^-j'^Oj whence tang.%}^=^j — J.' therefore the place where 



the reduction of latitude is a maximum, has the tang, of its latitude 

 equal to the ratio of the equatorial to the polar axis. It is worthy 

 of remark too, that when h is a maximum, the angle eCm, which is 



called the reduced latitude, has its tang, equal to -' as will readily 



appear by substituting the value of the tang, of the maximum value 

 of 5, in the relation between tang. C and tang. ■\', as found in 1. It 

 follows therefore that the reduced latitude, when a maximum, has 

 the value of its tang, expressed by the ratio of the polar to the equa- 

 torial axis. 



3. It is evident that all which has been said relative to the angle 

 made by the normal and radius at any point on the earth, will apply 

 with equal force to either of the other oblate planets; therefore formu- 

 la (6) may be applied to either of these planets by substituting the 



1 83 . 



proper value for a ; now, for Earth a=—. '■> for Mars «, = Tokc ' 



728 - 3 



for Jupiter a =T-T-j:^-' and for Saturn a.=^ . These values being 



successively placed in the maximum value of tang, h, we derive the 

 numbers in column 2. of the following table ; and these maxima 

 values being substituted, together with the corresponding values 

 of a in formula (6), will give the numbers recorded in column 3. 



Table of the maxima values of the Reduction of Latitude for the differ-ent, planets. 



The results expressed in this table give a very good idea of the 

 coKiparative degrees of flatness of the oblate planets, for the more the 



