84 ^^' Herschel on a remarkable Application 



f....p = ±(2L)".-v/T- • - • {8,5}. 



I n 



In fig. 2, let ASM be the axis of a parabola, BAG a tan- 

 gent at the vertex, S the focus — bisect the angle BAM by 

 AP, and draw /yn other lines AP, AP, AP, so that if 



1 2 3 M 



PA, PA, &c. be produced through A, this system of lines 



I 2 



shall make equal angles around A, then, neglecting the sign 

 AP. AP.. ... AP = (3L)". x/J. 



12 n 



If w = 4w -j- 3, the resulting value of p .... p is of the same 



I n 



form as 1 8,2 1 with the exception of a different sign. 

 If n be of the form ^m, or 4m -|- 2, 



2a" (I— x*)*" . sin. w^)/ - 



p P= + i r-^ .... ]8,^J, 



r '^ — 1 — 2^*". COS. 2«4/ + X*» (. ' ) ' 



where the sign +, or — , is to be used, according as n is of the 

 former, or latter form. 



When n = /^m + 2, if il; = — , this becomes 



p p=±^/-[Ti^: {«'7}. 



I 71 



and when x = 1. 



P /=±^' • • ■ {8.8}. 



in' 



Thus, in the last mentioned construction, if the number of 

 lines be 4W + 2, instead of ^m + 1 



AP . AP AP = <-^. 



2 



12 n 



The ninth of our primitive equations gives, if the angles 



