MI'.MMIK'S OK TIIK NA'I'loXAI. A(Al»i:.MV OT S( I IIN'CKS. 



Tliis cquiitinii is vmIIiI wliatover ho 'a , tlir iiiMJ.ir :i\is of the mliit (i\, nml m;iy Kr 

 mine the inii.jur axis cif cither orbit from (lie flcmeiits of the otiicr. My pri'sciil |. 

 over, to study tlie aelion of Jupiter in ehanf,'iii<,' ori)its tliat are (>ri;;iiially paralml;! 

 geuoral '*, will bo taken inlinite. In that ease 



use.! t< 

 rpi.se 



Am 



(-') 



It will he fouml that the seeoiul number of ('!) dipends on r.j, d and /;, and these are known 

 <iuanlities when theelenu'Uts of (!', and jf are <,'iven. 'riic use of the e(|uation is moreover K'^'iiHy 

 simplilied and eulianced by the fact that the plane id'tlie planet's orbit is involved only in so far 

 as that it must contain the tanfrent to j( at E. 



t>. In the second nu'mber of (-) all the factors are positive e.\ee])t eos </j, henee. if </^<i7r, Oi is 

 positive aiul the orbit f is an elli])se; but, if <p>i7r, 'w is negative and (f* is an hyjierbola. This 

 result may be thus expressed: \f the comet /xisiies in front of Jupiter the kinetic etierfii/ of the cornel in 

 tliinitii.ihed; if It ixixxex liehiml the plnnet Ihr kinetic enen/i/ of the comet is inerea-svil. The reason for 

 this nmy also be given in general language. If the eomet i)asses in front of the planet the (unnet's 

 attraction increases the velocity, aiul hence increases the kinetic energy of the planet and rice 

 rerun. But the total energy of the two bodies is constant; so that when that of the planet is in- 

 creased, that of the comet is diminished and rice versa. 



7. It is desirable now to transfoiin the value of -a given in equation (2) so as to l)e able to 

 determine the major axis of the new orbit of the eomet directly from the circumstances of its initial 

 approach to the planet before perturbation; in other words, to find 'a. in terms of oj, d and h. For 

 this we nnist find in terms of <<), d and h, values for s, p, a and (/)» 



.S. To find s. — In Fig. 1 let A and E represent the two points A and E as 

 defined above (Ai-t. 4), and the line AE represent d. Let AY be the tangent to 

 (t, at A, and EO the tangent to jf at E. It is an admissible .supposition that the 

 planet is describing the strtiir/ht line ()E, ami that the comet in its nnpei turbcd 

 orbit is describing the straight line YA. At some certain moment the line.j<iin. 

 ing the planet and the unperturbed comet must evidently be perpendicular to 

 ()E. Let OY be the line Joining the bodies at that moment, so that the planet 

 is at O when the comet is at Y, and EOY is a right angle. Instead, however, of 

 supposing the planet to move from O toward E we may apply an equal, opposite 

 motion to the comet, ami consider the ))la.uet to remain at rest at O. Draw AC 

 parallel to EO and make AB equal to the distance described by the planet during the time that 

 the comet is moving from Y to A. Join YB. Then since YA and BA represent in direction and 

 magnitude the motions of the two bodies in a given interval, the third side YB of the triangle rep. 

 resents in magnitude and direction the motion of the comet relative to the plaiu't. The angle 

 YAB is the angle w, and the three sides of the triangle YA, YB, and BA are ])ro[)ortional tor, Vq 

 and ?',. Let the angle YBG be H; then from the triangle YAB we have 



rQ^=vi^—2v,r cos cj+jj^, 

 and '•: (•,: rQ::sin 0: sin {B—o)): sin &;. (.3) 



Since r and p, can be com])nte(l from the given elements of the orbits of the iilaiiet and comet, we 

 nuiy readily compute from co the value of s, or i'^ v,. But if the jilanct is at its mean distance 

 from the sun, and the comet's orbit is parabolic, r- = 2r,', and we lia\ e 



X' = W - -ly/-' COS 0,5. (4) 



Also from the triangle 



lir/ = ('3- + 2ror,cos W+r/, 

 or 'Is cos W = 1 — .s--'. ^'5) 



!t. To find p. — The platiet being regarded at rest at O and the relatire un])erturbed nuition of 

 the comet l)eing along YB, this line may within admissible limits of error be treated as one a.symp- 

 tote of the relative orbit C. The )>eri)en(licnlar from O nixni YB will then be by definition (Art. 4) 

 the line p. Draw OX. from O yer|iendicular toOV and OI'^, and let these three lines be coordinate 



