60 Prof. Sylvester's Astronomical Prolusions. 



in the opposite branch, will be real or imaginary according as * 

 is greater or less thau c+4«, i.e. according as 2a is less or 



greater than ^-^. A third pair, in which the two given points 



are distributed between the two branches, will be real or imagi- 



c+ <r 

 nary respectively according as 2a is less or greater than ^ * 



and a fourth pair, where the same distribution occurs, will be real 



£—0* 



or imaginary according as 2a is greater or less than — ^— . 



It is of course only with the first kind of hyperbolae, that in 

 which the given points lie in the branch concave to the given 

 focus, with which the problem, regarded as an astronomical one, 

 is concerned. But in all cases the formulae given for the deter- 

 mination of e and i admit of immediate adaptation to logarithmic 

 computation. Thus, ex. gr., if we take the one which meets the 

 case of distribution between the two branches of an hyperbola, viz. 



„- l W~^ 



{<ro* + c 2 ) ± V (c 2 - a- 2 ) (c 2 -o^ 2 )' 

 writing 



e=sec<£, s=csec\, <ts=c cos fi, a 1 = c cos p} , 



we obtain 



tan (/> = tan \ sec 



± cos i= cos ^ cos <f>. 



Viewed as a question of analytical geometry, the investigation 

 as to the reality of the curve would have to be treated in a more 

 general manner; i. e. without assuming, as I have done, the neces- 

 sity of the inequalities^ > p x + p^c >p x — p%, wherep^^represent 

 the two given focal distances ; for it is a very important, although 

 hitherto strangely neglected geometrical principle, that every 

 real conic is at one and the same time an ellipse and hyperbola ; 

 viz. either an actual ellipse accompanied by an ideal hyperbola, 

 or an actual hyperbola accompanied by an ideal ellipse. This 

 may immediately be made manifest by considering how the ordi- 

 nary rectangular-coordinate equation to a conic, with its origin 

 transferred to a focus, is connected with the property of the 

 conic in respect to its two foci. Calling p, p 1 the two focal dis- 

 tances of any point, the equation to rectangular coordinates is 

 obtainable by equating to zero the norm of the quantity 2a ±p ± g 1 , 

 where p represents Va? + y 2 , and pf represents V (2ap + xf- + y% 

 which norm will only be of the second degree in x, y, although a 

 product of four factors each of the first degree in x, y, owing to 



