12 Mr. Herschel on a remarkable Application 



it will then be free from imaginary symbols ; thus 



4 .sin. -^ . sin. 



r r^ == 



I n 





2 2 



When ^ = 1, or the conic section is a parabola, x= i, and 

 we find 



f f'=qzS:;ij="^)"-^*— ©'f-^i- 



A result of such remarkable simplicity, as deserves a more 

 particular enunciation. Let then, in the diagram, fig. i, S re- 

 present the focus of a parabola A,P,Q, and, having drawn any 

 line SP, make n angles PSP, PSP PSP, about S, all equal 



I 1223 71 I 



to each other; draw the axis ASM, and make the angle 



MSQ = n times MSP; and if L represent the latus rectum, 



I 

 we shall have 



SP.SP SP—L^'-'.SQ, 



«— I 



12 n 



for, by the polar equation of the curve, SQ = -^ . cosec. [ — 54*. 



Thus, if SP be coincident with SA, and n be odd, cosec. — = 1, 

 I 



andSP....SP=-;-L". . .U,A. 



nO 



but, if SP be perpendicular to SA, and n still odd, cosec. ~ = 

 I 



v/7.andSP....SP = yL'* ji,5|. 



If SP be perpendicular to SA, but 71 of the form, 4/;^ + s 

 I 



sp....sp = -iL" \i,e]. 



in* ••J 



