.Mi;.M(lll;S OK TIIK NATIONAI. ACAIHIMV OF SCIKNC'KS. 



I'roiii (7) 



iM-diM I Id) and (S) 



A-'+H' 



I J' 



(•(IS / sin " I cot / Hill 



(I + s\u' I UurH)^ (sec^ (9+cot»/)* ' 



From these ainl(!t) 

 and liciicc tViiiii (L') 



sin I cos \ = 4- '' '^'"^ ^ = + '*^'*i°i^ 

 (if'+h' sill' f))^ ^ 



OS </' sin ,. (A- + B2) = AB cos H ,1: lili sin' ft, 

 im • A cos H 1- /( sin- W 4»» * Acos/V ± /i siii'^- 



(13) 



Since m is the known mass of the j)laiiet, and w, .v and A are known functions of w, equation (Ui) 

 jrives directly tlie vahie of®, the seiuiaxis major of the new orbit (f in terms off/, h and c.i. 



14. For a particular case of approach, equation (13) is convenient for computati(jn. We may, 

 however, now treat d, h and co as independent variables whose varying values may express all the 

 different possible cases of approach of the comet to the i)lanet so far as change of ])eriodic tiuii^ of 

 the cc met is concerned. The dependence of <«) ui)on the three variables can not be very easily 

 represented graphically in a single plane diagram. But by giving to co successive values in mul 

 tiples of 10=', viz, 0=10°, 20=, 30°, etc., to 17<)o, I have prepared a series of diagrams to exhibit 

 in eaitli case in succession the relation of ® to the other two variables. The values of d, « and A 

 for the several values of &> were needed in making the diagrams, and they are given in Table 1. 

 l"](juations (4), (5), and (7) are used in making the table. The disturbing ])laiiet is assumed to be 

 Jupiter, so that m was taken ccjual to 1 1050 and r='r2. 



Table I. 



15. Using these values of 6, s and A we may now represent graphically the dependence of 'S 

 upon the other two variables d and h for each specified value of co. Let d and h be Cartesian coor 

 dinates, then for each point of the coordinate plane there is a value of ®. The ambiguous sign 

 will be fully satisfied by giving positive and negative values to /(. For an assumed value of <<< 

 we shall have a curve whose equation is (13), and each |»oint of this curve represents values of d 

 and /( for which the total action of tlie planet upon the comet will be to reduce the energy of tlie 

 comet a constant anioiuit. This locus will be called an isergonal curve. 



l(i. Fniscenii <>/ ixergoiml ellipses. — The equation (13) of the isei'goual curve may be written 

 4wfe (A cos i9+/t sin' ^)=s(A'+d'+/i' sin' (9), 



and this is the equation of an ellipse. As ® changes its value we may treat it as a parameter and 

 we have a faisceau of similar isergonal ellipses, e-ich ellipse symmetrical with tlie axis of h. The 

 radical axis of the fais(;eau Acos«+/(siii-''=(l, and the imaginary ellipse A- + </- + /(-siu- W=0, 



