Olbers on Comets. 155 



X b — a y _ h — a a + b — \^({a\- bf — c^ 



2p— y Ap" y ^" yy 



= ^ *- ■ ^ - ^ . It IS obvious that rp re- 



y . . ^p 



mains constant as long as a or r' is one of the distances com- 

 pared.] 



The two remaining equations are found by a comparison of 

 the chords and distances from the sun with the observed in- 

 tervals ; whence we have 



'r'4- r" + A'\| (r' + r" — k'\% 



t' — 



LU_(!i 



m3 ^2 



_ / r -f- r" +^" \| /r' -\-r"'—Ji' \i 



'^ "^ V ^ / \ ^ / " 



»i3 -v/2 

 m denoting the quantity employed in the same manner by Euler 

 and Lambert. I am not aware that these four equations have 

 before been exhibited in this manner in their simplest form. 

 [Note 4. C. The area of the sector, comprehended between 



p r - ^1 



the distances a and b, is Vrr L(^ + '^ + ^)- — (a -h b — c)" J 



The area of the triangle contained Ijy a, b, and c is ^ y' 

 {(a-\-b + c) (a + b — c) (a—b + c) f— a + 6 + cj] = i 



V ([a + hY — c"; V (c' -(h — af) -\^ {\_a -^ bj — c), 



and that of the. remaining segment of the sector is equal 



to the area of a segment at the apex, of the same depth, 



?/?/ ?/' y 



and of the same breadth, ?/, or to f y , -—r- "^^ t. = ';: 



•" •* ^ 16p 24j9 6 



(a + /j — V C[a -h lif — c-) ; the sum being ^L(a-\-b)-\-^ 



•sf (\a\b'^ — tV , of which the square is ~- [2 f « + h) 



1 4 



+ V /T^' + '-'? — cV IS and this multiplied by - zz — 



v yy 



[ a + /; - V r [« + bY — cV ] gives ^'^ (2e+f )'(e —fj 

 calling a+ b, c, and >/ f [« + 6]'' — c'J, / , or vj'g (4 e' — 



