154 Astiviiomieal and Nautical Cu/lcctious. 



(x"—x') = — x' z" + z' x" to y' (z' — z")—z {y' — y") = — 



7j' z" + z' y , or as X z — x z to y z — y z ; and - _ 



y ^ ^-f which must also be equal to , ■ ,„ H;; — >, sub- 



x' z" — x" z ^ x z —X z' 



stituting y'" and z" for j/" and z" ; and the fractions will remain 

 equal if we refer the angle to a plane perpendicular to the 

 ecliptic, and employ y', y", and y" in the denominators, instead 

 of z', z", and z". Tr.] 



The second equation depends on the condition, that the three 

 places of the comet must be in a parabola, of which the sun 

 occupies the focus : hence 



— 2r'+V ([r'+ rJ — Jr- )_ —Ir'+Vgr'+r'J — k"') 

 ^ (U"- — [r" _ r'Y) V r^" ' — (r'" — r) '). ' 



[Note 3. A. If two distances of a comet from the sun be a and 

 h and the intervening chord c, the perihelion distance p ivill be 

 _ ^ c^_(b — a)' 



- «■ a + b — V ([a + bp— c') " 

 The ordinates perpendicular to the axis at the respective 



points being x and x + y, we have a—p -\- r^, b — p -\- 



(x 4- 7/ Y 



* — •''' , and 7f — c' — (b — af; consequently b—a — 

 4 p 



Ixy+yy x _ h —a V xx __ (b — a)" yy 



^ ' '2p~ y 4p Ajjp" yy 16pp 



b — a _a_ (h — uf- ^ _ (b — af^- c- — (b — af 



cc , cc ?/y i + o 1 B(a + h) _ 

 — __ and — -L- •'■ ■ ■ zz —: ; whence _ — 



~yy' yy ^Q PP ^^P pp Pyy 



16 cc 1 a + b _ / 16 (a + bf _ I6cc 1 _ ^ 



~V 'p~ "^ yy ~ \ y' f ' p~ ' 



a + bt^s/([a + bY — c') ^^^^ conditions of the problem re- 



c'—(b—af 

 quiring the lower sign, 



B. Employiwj the same notation, ice find ~ — = 



2 a— v/('[a + b]- — c'1) the first ordinate. 



