VOL. LVII.] PHILOSOPHICAL TRANSACTIONS, 878 



never suspected that it would affect the durations of the eclipses of the satellites, 

 till Dr. Bevis first thought of it, in the latter end of the summer 1761. The 

 Doctor, being at that time indisposed, recommended the subject to Mr. W.'s 

 consideration ; and in consequence Mr. W. not long after presented him with a 

 solution of the problem, being in substance the same with this, as far as propo- 

 sition 5 ; a copy of which he soon after transmitted to that excellent mathema- 

 tician the late M. Clairaut. 



In March 1763, M. de la Lande, an eminent French astronomer, being here. 

 Dr. Bevis showed him Mr. W.'s paper ; this occasioned a new article in the 

 Conn, des Mouv. Celest. 1765, p. 177, under the title, Inegalitd dans les demi- 

 durees des eclipses des satellites de Jupiter, causee par I'applatissement de Jupiter: 

 in which he mentions this circumstance in the following words ; M. le Docteur 

 Bevis me fit voir k Londres, au mois de Mars dernier, une solution rigoureuse et 

 algebraique de ce probleme, qui consiste a trouver la courbe qui resulte de la sec- 

 tion de I'ombre d'un spheroide a une distance quelconque. 



A few months since, M. Bailly, a French gentleman, published at Paris an 

 elaborate treatise on the theory of Jupiter's satellites; in which he has been 

 pleased to give the honour of this discovery entirely to M. de la Lande, without 

 the least mention of Dr. Bevis. Mr. W. then thought it incumbent on him to 

 do justice to the doctor, by immediately finishing his paper in the best manner 

 he was able, and presenting it to the Royal Society. 



Lemma. If any spheroid be cut by a plane, in any direction whatever (except- 

 ing that which is perpendicular to its axis), the figure of the section will be an 

 ellipsis. This is demonstrated in Simpson's Fluxions, vol. 2, p. 456. 



Prop. 1 . In pi. 8, fig. 9, Let the sphere begk be cut through its centre by 

 the planes bgk, bpd, bod, bod, eak, and lph; it is required to determine the 

 inclination of the planes lph, bod, and also the inclination of the right lines AC, 

 BC, which is measured by the arc ab; there being given the angles of inclination 

 EBF, Fha, together with the arc bf: the angles afb, eal, being right angles, 

 and the inclination of the required plane bod, but little exceeding that of the 

 given plane bod. 



Let the sine of ebp = a, its cosine = a, the tangent of bf = b, its cosine 

 = b', the sine of pb« = p, its cosine = p', the sine of aba = z, the sine of ab 

 = z, its cosine = z', the sine of pao = ^, and radius = 1 ; then will the sine 

 of ABF = the sine of (fba -}- oba) = p -\- p'z, and its cosine = p' — pz: there- 

 fore by trigonometry, in the right angled spherical triangle abf, as rad. (1): co- 

 sine BF {b') :: sine abf (/> -|- p'z): cosine baf = sine of lab, or its equal pao; 

 therefore ^ = i' X (/> + p'z) = the sine of the required inclination of the planes 

 LPH, bod. In like manner in the same triangle it will be as rad. (1): cotan bf 



