382 PROCEEDINGS OP THE AMERICAN ACADEMY 



cos FB = COS BC COS CF = COS CA, 

 .*, C03CA< COS BC, .*. CA> BC. 



14. The equation of the conic may be determined by finding the 

 locus of the vertex of a triangle, when the base and the sum of the 

 sides are given. 



Suppose A -\- B = 2a ; then 



COS A cos B — sin ^ sin B = cos 2«, 

 (cos ^ cos ^ — cos 2ay = (1 — cos'-' A){1 — cos'^ B), 

 cos'' A -j- cos- B — 2 cos 2« cos A cos B = sin'^ 2a ; 

 but cos 2« = 1 — 2 sin^ «, sin 2a = 2 sin a cos a ; 



. •. (cos A — cos By -\- 4 sin^ a cos A cos B = A sin^ a cos- a. 

 Let FC = c, and use the formulas of Art. 3 ; then 

 x^ ian^ c-\- {I — x^ tan^ c) sin^ a := (1 -\- x- -\- y") sin^ « cos^ a sec" c. 



But cos c 



cos a 



cos /?' 



cos- /3 g 



^'^'^ '' = Zi^^a^ tan-c = 



C0S2 /3 — CDS'* O 



and hence, by easy reductions, 



x^ cos^ « sin'' ^ -{- y " sin'' « cos- (3 = sin" a sin" ^, 

 or a;" cot" a -|- y" cot" ^ r= 1, 



which is the same form of equation as that given in Art. 5. 



15. If the sphere be so divided as to make a spherical hyperbola, 

 then the locus becomes the vertex of a tx'iangle whose base and also 

 the difference of its sides are given. 



f"p -}- pp = 180°, f'p -|- pf = constant ; 

 , • . f"p — f'p = constant. 



