of the Second Degree. 77 



cd- + ae- — bde _ (*J c d— *Ja e) s 



X J a er=- 



sin-0 sin-0 



whence the equation becomes 



«J c d— «J a e 



y~± ~, 7= r— — jSirr0:r = o. 



(a — 2 y/ a c cos 6 + c) i 



The position of the axis is determined by the same criterion 

 as that for finding the major axis of the ellipse; and the curve 

 may be further ascertained if we recollect that it must lie en- 

 tirely on one side of the right line whose equation is dy + 

 ex+f-O, and on that side in which the co-ordinates of a point 

 are such as to make dy + ex+f of a different sign from a or c. 



In order to find the co-ordinates of the vertex, we must have 

 recourse to equations (5) recollecting that B--aAC=o, and that 

 tan <p tan f + 1 = 0. From the third and fourth of these we find 

 E _ _ </A__ _ N /fl- N /ccosfl 



But the fourth and fifth of (4) give the following equation 



D _ 2 s/a k + d- cos 9 (2 ^fc k + e ) 

 E i'-ty/c k + e) sin0 



where k = +fan + «Jc m 



and the last of (4) gives 



k 2 + dn + em +f= 0, 



from the two first of which we find 



a ~ 2 */ac cos + c 

 and from the last 



dn + em= —f—k", 



from which equations m and n may be found. The expressions 

 present nothing remarkable. 



