146 PHILOSOPHICAL TRANSACTIONS. [ANNO 1781. 



of the Astronomy of Comets, to illustrate the mode of computation that he 

 pursued in constructing his general table for calculating the place of a comet in 

 a parabolic orbit; and it is obvious, a being put for the known side of the equa- 

 tion, that it may be transformed to \/ x X \/(3 + x 2 ) = \/ a: where, if x be 

 considered as the tangent of an arc, the radius of which is \/3, then \/(3 -\- x 2 ) 

 will be the secant of that arc; and consequently, by what is shown in the 1st 

 example, an arc must be found such, that the sum of the tabular log. secant and 

 half the tabular log. tangent may be equal to the excess of half the log. of a 

 above f of the log. of 3. In the first of the above 3 instances this excess will 

 be found, 18.943 1891, in the 2d 19.0937041, and in the 3d, 19.1817497; and 

 by running the eye along Gardiner's tables of logarithmic sines and tangents, it 

 will be found, that the first falls between 0° 26' 20" and 0° 26' 30'', the 2d be- 

 tween 0° 52' 50" and 0° 53' 0", and the 3d between 1° 19' 20" and 1° \g' 30"; 

 and, by pursuing the mode described in the former 2 examples, the exact arcs 

 will be found 0° 26 27"7, 0° 54' 5l"7, and 1° 19' 20".l, and their respective 

 tangents, to the radius y/3, .01333248, .02666 1 1, and .0399787, the 3 values 

 of x sought. And in this manner Dr. Halley's table may be extended to any 

 length with the utmost ease, expedition, and accuracy. 



Thus far this matter has been carried by former writers; but those who may 

 be at the trouble of consulting them will find that I have not copied their me- 

 thods: on the contrary, these which are given here are more plain and obvious 

 than theirs are, and the operations considerably shorter. What follows has not, 

 I believe, been adverted to by any before me. 



Exam. 4. — Let the equation arising from the proportion a :b -f- x (l — c 2 ) :: 

 c </ (l — x~) "• c^x be taken, which is the result of an inquiry into the situation 

 of that place on the surface of the earth, considered as a spheroid, which is at 

 the greatest distance from a given one, suppose London. In this inquiry a and 

 b were put to represent the sine and cosine of the latitude of the given place, in 

 the spheroid; c for •£-§-§-, the ratio of the axes; and x for the sine of the dis- 

 tance of the required place from the opposite pole, in the spheroid also. The 

 equation, which is of 4 dimensions with all the terms, is manifestly acx = 

 [b +x(l — c'")] X V {\ — x 2 ), or — — — - — x . — ^— = -; in which it is 

 evident from the tables that the difference between the tangent and the product 

 of the sine into a given quantity is known. In order therefore to find the value 

 of x, compute — , and , and find the logarithm of the latter. Now because 



the elliptic meridian differs but little from a circle, the place sought will not be 

 far from the antipodes of the given one, and its distance from the opposite pole 

 may therefore be estimated at 39° 5 ; and, having taken out the natural tangent, 

 and logarithmic sine of this arc, add the logarithm of ■ ~ ' ■ to the latter, and find 



