10 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



axes. Let the line AB meet the plane XOY in C. Join OC, let fall OD perpendicular to YB, and 

 join CD. Since EA is perpendicular to AY and also to EO, and so to its parallel line AC, there- 

 fore it is perpendicular to the plane YAC. Hence OC, parallel to EA is perpendicular to the 

 plane, and so perpendicular to CD. Again, CDY is a right angle; for OD2+DY2=OY2=OC2+CY^ 

 and OD2r=OC'+DC^. Hence, DC^+DY2=CY^ and consequently CDY is a right angle. 



The quantity /( is the line BC ; for h is the distance which the planet, when the comet is at A, 

 has yet to pass over before reaching E. But the cornet was at Y when the planet was at O, and 

 the planet describes BA, while the comet describes YA, leaving BC as the distance yet to be de- 

 scribed, or h. But the angle CBD is 0, so that we have 



7/ = OD2 = OC^ + CD2 = d' + h^ siu^ 0. (6) 



10. To find a. — The angle n is the acute angle between the asymptote and the transverse axis 

 of the hyperbola, and hence from the nature of the hyperbola tan <>=B/A. By known formulas 

 we have, if the planet is at its mean distance 



Therefore 

 Hence from (6) 



■■d)i 



(7) 



11. To find cf>. — The orbit o&the comet relative to Jupiter lies in the plane YOB. Leti be the 

 inclination of the plane YOB to YOX, measured positive from x positive to ;: positive; let I be the 

 longitude of the direction YC, measured in the plane YOX from OY, that is, the angle made by YC 

 with OY produced; let A be the longitude of the direction YB measured in the plane YOB from 

 OY, that is, the angle made by YB with OY produced. Imagine now a sphere described about 

 Y as a center that shall cut the three planes XOY, BOY and BCY in three sides of a right-angled 

 spherical triangle. The hypothenuse of this triangle is A, the base i, the perpendicular i^n —(9, and 

 the angle opposite to the perpendicular is i; hence we have 



cos A = cos I sin (9, (8) 



cos 8 = sin i sin A, (9) 



cot i = sin / tan (i. (10) 



Also from tlie triangles OCY and BCY 



tan I = tan OYC = - JJ^ :z= - r^^-^ (11) 



YC /( tan 0. ^ ' 



The angle tp is by definition the angle between the direction OE, and a line in the i)lane YOB that 



makes with YB an angle «. Hence we have readily 



cos (p = sin i sin (A ± a). (12) 



These equations enable us to compute <^ in terms of d, h and on; for in succession O may be 



computed by (3), I by (11), A by (8), i by (10), and <p by (12). 



12. These values of s, p, x and a give by equation (2) the value of ® . The suppositions that the 

 planet is at its mean distance, and that QI, is a parabola, are involved in that equation, but they 

 are not necessary to the determination of® when no siudi hypotheses are made, and cluuiges in 

 the equation that are not serious would make it applicable without these limitations. The quan- 

 tities in the several equations may be regarded as having values: — 



d positive, 



h positive or negative, 

 a xjositive and less than ^ n, 

 CO, 0, q) and i positive and less than n, 

 I and A positive and less than 2 tt. 



1;?. W(- may, however, also find directly the value of ■?/ in terms of d, h, and the known func- 

 tions of u\ 



From (12) 



cos <7J sin « = sin i sin A cos a sin a rt sin i cos A sin^ a. 



