Prof. Sylvester's Astronomical Prolusions, 57 



f i\ +/ 7T V +sc— \/s—c 



m (q + q')=:\/s+ //A = 



2 

 =- \/s + c+ a/s— c 



Vs + C . */s — 



/— 



mq= — — , wig' = — - — ; 



and thus 



fe^(#+c)t-"(t--e)f). 



There is a sort of pendant] to Lambert's theorem which is 

 worthy of notice. If we call p—p'=v and c 2 — <7 2 =D, writing 



«e (sin w'-— sin u) = H, 

 we have also 



(1 - eV(l - cos(a'- w)) =D, 

 £« (cos u— cos w') = a j 

 from which we easily obtain 



«-^ 



so that 12 is a function only of c, e, <r, as by Lambert's theorem 

 it is a function only of c, a, s. Moreover, since 



■K^)-\4£ 



2(1-^) 



it is apparent that the change in the mean anomaly is a complete 



\/D c 

 function of the two variables — = — , -, as by Lambert's theorem 



k/Ac h ." 



it is of the two > -. Comparing the value of XI given im- 

 mediately above with that which is contained in Lambert's 

 theorem, the solution of a linear equation leads immediately, 

 after certain simple reductions, to the equation 



, . 2(c*-<r*) 



6 " ss 1 + c 2 + 4/(s 2 -c 2 ) (s^-c 2 ) ' 



where s' + s=4«. And as there is nothing to determine the 

 signs of p or p', the above, by interchanging severally and inde- 

 pendently p } p ! with — p, — p\ represents eight values of e: — four 

 corresponding to the change of p into —p, and p' into — p', con- 

 tained in the expression immediately above written, combined 

 with the equation s , + 5=+4a ; and four in the conjugate ex- 



