Mr. Ivory's Correction in his Geodetical Problem. 249 



figure of the planets, the principles of equilibrium laid down 

 by Clairaut are supposed to be sufficient ; and the same ob- 

 jections we have already noticed in the investigation of Le- 

 gendre are equally applicable to the researches of Laplace. 

 The speculations of this great geometer mai'k the limit of our 

 attainments in one branch of the philosophy of Newton : on 

 this account they are deserving of a more ample discussion 

 than we can at present bestow upon them, and they will pro- 

 bably engage our further attention at some future time. 

 April 3, 1825. James Ivory. 



XL. Correction of an Inadvertence in a Geodetical Problem 

 inserted in the Philosophical Magazine for Jtdy last. By 

 J. Ivory, Esq. M.J. F.R.S. 



To the Editor of the Philosophical Magazine and Journal. 

 Sir, 

 T N the geodetical problem inserted in the Philosophical Ma- 

 •*■ gazine for July last, I find there is an inadvertency with 

 regard to the designations of the angles v|/ and A, which are 

 not the true latitudes, but angles having a very simple rela- 

 tion to the latitudes. But this circumstance affects neither the 

 reasoning nor the accuracy of the formulae, which are strictly 

 true when the proper values of the angles \J/ and K are under- 

 stood. 



Conceive a semimeridian upon a diameter of the equator, 

 and likewise a semicircle upon the same diameter ; and from 

 a point in the diameter produced draw tangents to both curves: 

 then, the angle X is the inclination of the tangent of the circle 

 to the polar axis ; and the true latitude, for which I shall put 

 /, is the inclination of the tangent of the ellipse to the same 

 axis. The angle X is by some authors termed the reduced or 

 corrected latitude ; and my solution is true and correct if the 

 reader will be pleased to observe that \J/ and \ stand for the 

 retluced latitudes, and not the true latitudes. 



The property of the ellipse furnishes this equation, 

 tan I 

 tan K = ,- : 



sin / 



whence we get sin K = / . '. ==■ 



, cos / -v/l+c- 

 COS X = — 



Vl + c- cos-Jf 



and, very nearly, x = / — ^- sin 2 /. 



Vol. 65. No. 321. April 1825. I i These 



