62 Prof. Sylvester's Astronomical Prolusions. 



The complete arithmetical determination of the signs to be 

 given to the several quantities a, /£, y, 8 requires a distinct and 

 detailed examination, which it would be irrelevant to enter upon 

 in this place ; it is enough to see that a second focus G at the dis- 



tance t from a given one F may be moved along the line FG to 



•y b 



a new focus H at the distance ^ from F, the modulus - becoming 



R it 



simultaneously replaced by -, and the constant - by the constant 



-. I am not aware that M. Chasles has ever disclosed the 

 a 



aperqu which led him to this unlooked for discovery. It is to 



be hoped that he will not allow future ages to labour under the 



same doubt as to the source from which he drew it as we must, 



it is to be feared, ever remain under with regard to the origin of 



Newton's rule, recently demonstrated, or Lambert's theorem, 



the motive to this paper. In this age of the world euristic is 



even more important to the promotion of science, and its secrets 



less likely to be recovered than those of mere apodictic. 



Since a focus may be regarded as the intersection of two tan- 

 gents at the circular points of infinity, we may generalize the 

 problem of constructing the orbit by considering it as a particular 

 case of constructing the conic which passes through two given 

 points, touches two given straight lines, and has a principal axis 

 of given length. 



Taking the two given lines supposed to be inclined to each 

 other at the angle a as the axes of coordinates, the equation to 

 the curve may be written under the form 



(Aa? + Cy + I) 2 =2B 2 #y, 

 which, writing 



-*_ c _ 1 A 



*~* 2AC-B 2 ' y ~ v 2AC-B 2 ' 

 becomes 



A«p+2(A0-B^f,+G,«=gj^^. 



Adding \(f*— 2 cos a £77 4- ?7 2 ) to the left-hand side of the equa- 

 tion, the discriminant of that side so augmented becomes 



(sin a) 2 \ 2 +(A 2 + 2cos «AC + C 2 -2cos aB 2 )X + B 2 (B 2 ~2AC). 



Hence, calling the squared reciprocal of the given principal semiaxis 



2B 2 

 q, and writing X= j— ^ — tT 9j we obtain 



4sin «VB 2 + 2(2AC-B) 2 (A 2 + 2cos «. AC + C 2 -2cos« . W)q 

 + (B 2 -2AC) 3 =0; 



