es P . 
232 Lo 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. é,equal to 
zero, and the resulting value for tang. 1, 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—a)? X 
tang.?.)=0, whence tang-+=7_, =33 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 4 is a maximum, the angle eCm, which is 
b 
called the reduced latitude, has its tang. equal to 7 38 will readily 
appear by substituting the value of the tang. of the maximum value 
of 6, in the relation between tang. C and tang. J, as foundin1. 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 
] 8 Bd 
proper value for « ; now, for Earth a=354 > for Mars « = ja553 
) 728 3 
for Jupiter “=70,000" and for Saturn c=5,. These values being 
35° 
successively placed in the maximum value of tang. 6, we derive the 
numbers in column 2. of the following table ; and these maxima 
values being substituted, together with the corresponding values 
of « in formula (6), will give the numbers recorded in column 3. 
Table of the maxima values of the Reduction of Latitude for the different planets. 
i ee Fa ie eee 
1 
"f 2. : 3. 
_Names of Planets, _ Maximum value of §. Latitude where § is a maximum. 
Jupiter, = - 
Saturn, 
The results expressed in this table give a very good idea of the 
comparative degrees of flatness of the oblate planets, for the more the 
| 5° 07’ 39.14768” 
Earth, - 0° 11 26.34204” 45° 05’ 16.80315/ 
‘Mars, . 3° 37 09.52994” 46° 49’ 34.71429” 
4° 19 36.50257” 47° 09/ 47.92381” 
RE 00' 40.0980 5 
