VOL. LVII.J PHILOSOPHICAL TRANSACTIONS. 377 



It remains now to apply what has been investigated above, to the eclipses of 

 Jupiter's satellites, and to examine whether the prolateness of his figure will have 

 any sensible effect on their durations; and this is become the more necessary, as 

 that celebrated astronomer M. de la Lande (who candidly acknowledges that he 

 was excited to turn his thoughts on this subject, from a cursory view of this 

 paper, which was shown him by Dr. Bevis) does not seem to have considered 

 the question with that degree of attention which Mr. W. thinks it demands. 



But before this can be done with exactness, it will be necessary to have the in- 

 clination of Jupiter's axis, with respect to his ecliptic, and the place of his equi- 

 noxes, determined by observation, neither of which he believes has yet been 

 done with any degree of certainty; he therefore proceeds in this inquiry on M. 

 de la Lande's hypothesis, that Jupiter's axis is perpendicular to his orbit; and 

 perhaps this supposition is not so far distant from the truth, as to occasion any 

 material error in the conclusion. It may also be remarked, that in the general 

 equation given above, v' and v express the sine and cosine of the semiangle of 

 the cone of Jupiter's shadow; but this angle can never exceed 3', and conse- 

 quently we may very safely use the radius instead of v wherever it occurs. 



By this means the general equation will become r — dv' = y^fTI^, or, which 



is the same, r — dv' = x, therefore v' = —j-i but by prop. 4, v' = "XZ^V 



which, because q is nearly equal to v, and q'\j very small with respect to gA, will 



become v' = — -— ; therefore ^—r— = ^-^, from which we shall find u = ~7-^; 

 a a A ' a 



and this equation is exactly the same with that which would arise from considering 

 the sun as a circular, and Jupiter as an elliptic plane, limited by one of his me- 

 ridians, and always parallel to the disk of the sun ; which supposition, the im- 

 mense distance of Jupiter from the sun renders very allowable. 



From this equation an easy mechanical method may be derived of delineating 

 the curve of the shadow, at any given distance from Jupiter. For as x denotes 

 any semidiameter of the elliptic section of Jupiter's body, it is manifest that the 

 term ^ X x vvill express the corresponding semidiameter of a similar ellipsis, 

 whose axes are to those of Jupiter, in the given ratio of A to d, and the term 

 J is wholly given; therefore if arm (fig. 15) be such an ellipsis, and there be 

 drawn through its centre m any number of semidiameters Ma, ub, mc, &c. meet- 

 ing the ellipsis in a, b, c, &c. let ax, bs, cc, &c. be taken each equal to the 

 given term -j, and the points a, b, c, &c. will be in the required curve. 



It appears, from considering the nature of this curve, that it will have two 

 cusps, one at each extremity of its lesser axis, which will approach toward each 

 other, according as the distance i is augmented; therefore, if the distance of the 

 section of the shadow, from Jupiter's centre, was taken such, that i = 



VOL. XII. 3 C 



