516 REPORT — 1891. 



determination of @ when no such hypotheses are made, and changes in 

 the equation that are not serious would make it applicable without these 

 limitations. The quantities in the several equations may be regarded as 

 having values :— 



d positive, 



h positive or negative, 

 a positive and less than ^, 

 to, 6, (f}, and i positive and less than tt, 

 I and A positive and less than 27r. 



13. We may, however, also find directly the value of @ in terms of 

 d, h, and the known functions of w. 



From (12) : 



cos sin a = sin i sin A cos a sin a + sin i cos A sin^ a. 



From (7) : 



AB , . , W 



cos a sm o = -— — — —, and sin'' a = 



From (10) and (8) : 



X i_ cos I sin _, cot I sin 6 



sm i cos A =; ± , = ± 1 , 



(l + sm^ /tan2 Oy- (sec^^ + cot^ ly 



hence from (6) and (11) : 



• . , h sin2 6 h sin2 



sin I cos A = -+- = -). ^ 



— (d^ + h'^ sin^ ey — B 



From these and (9) : 



cos ^ sin a (A^ + B2) = AB cos 6 ± hB sin^ 0, 



and hence from (2) : 



@= -^ . A^ + B'' ^ j_ A+J-^ +h^ sin2 6 



4:111 A cos 6 ± h sin- $ 4m ' A cos ±h sin- 6 ' 



Since m is the known mass of the planet, and 0, .«, and A are known 

 functions of w, equation (13) gives directly the value of @, the serai- 

 axis major of the new orbit C, in terms of d, h, and <o. 



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

 computation. We may, however, now treat d, h, ana w as independent 

 variables whose varying values may express all the diS'erent possible 

 cases of approach of the comet to the planet, so far as change of periodic 

 time of the comet is concerned. The dependence of @ upon the three 

 variables cannot be very easily represented graphically in a single plane 

 diagram. But by giving to co successive values in multiples of 10°, viz., 

 to = 10°, 20°, 30°, &c., to 170°, I have prepared a series of diagrams to 

 exhibit in each case in succession the relation of @ to the other two 

 variables. The values of 6, s, and A for the several values of w were 

 needed in making the diagrams, and they are given in Table 1. Equa- 

 tions (-l-), (5), and (7) are used in making the table. The disturbing 

 planet is assumed to be Jupiter, so that -ju was taken equal to 1/1060 

 and r = 5'2. 



