56 



Prof. Sylvester's Astronomical Prolusions. 



Such coincidence, to a mind open to the impressions of analogy, 

 could hardly fail to suggest the necessity of the existence of 

 a deeper-seated law, of which these extreme cases must repre- 

 sent the outcroppings. Euler's theorem is of course included 

 as a particular case in Lambert's, and may be derived from it by 

 making a infinite in the expression for t as a function of s, c, a; 

 but it may also be obtained independently as follows. Calling 

 4»i the latus rectum, retaining the rest of the notation, and 



writing tan ^ = q, tan -= = q 1 , we easily find 

 A A 



±\/A=m{l + qq'), 



s = m{2 + q* + q% 



Hence the Jacobian 



Hi^/A, s, 3. t) 

 d i m > 9> Q 1 ) 

 becomes a multiple of the determinant, 



Calling this 



it will be found that 



(A; B; C)- 2CT (D; E; I , )+^_(G;H;K)=0;-B; 



{Aj Bj C > + 3a|i (D '' E; F )"2AT1 (GJ H; K) = ° J _C;C; 



and consequently the Jacobian in question, as before, takes the 

 form 



1 + qq r ; q< ; 



2 + q* + q'*; 2q; 

 3( q -q') + q3-q«;2 + 2q*; 



2q' 

 -2-2q* 





ABC 

 DEE 

 G H K 







A; 



B; 



C 



0; 



B; 



-B 



0; 



-C; 



C 



which is identically zero ; so that t is a function only of s, c 

 when a is given, and one solution is left free between m, q, q r . 

 Making q— x , we have 



