Astronomical and Nautical Collections. 355 



Then, observing all the signs, from these values of s, s, s, . . ., s, s", 



s'". . ., r, r', r". . . and r, r, r, ... we find for the angle FHG, 



1 • •• \ z 



( t=, (rs — r's + r"s — . .) the value (cy 



^ 2c + 2cd 2c + 2cd^ "^ X 



— c^y ^ + c'i' -—- - c'y . + - — A]...) : and 



or X L ^ ozz 



for the angle FGH, (= {- rs' + r s" — . . — P) the 



2c + 2cd 



value (yx + .£- + A + 



2c + 2crf '^ c Ll-3z 5 z' 7 2^ 



5 +...]+ ^ r ^' + — - — 4 + Z_i ^ B +. . .] 

 -■ M Ll.3.5r 7 Z-" 9 z' ^ 



— P) : the correction P being the value of the same series for the 

 point A. 



Another series may also be found for the angle FGH, by cor- 

 recting the fluents r, r', r", so that they may all vanish in the point 

 A, when z ir a. For this purpose we may make z s= a — v, 



whence z zz — v, and the fluxion of the angle FHG is ^y^ . 



X 



1 z ' ' 



. Putting therefore, as before, s = — , we have r = vy, 



2c + 2cd 



w being — — v, and y =: — ~ ^'Z/ ; whence r ==: — cy, and r=: 



c 



cd - 01/, d being the value of ?/ at A ; consequently nw = —cdv + 

 cy'v = —cdv— c^y, and r = c^d—cdv—c^y; hence r" = c^d—c'dv + 



}.cdv^—d'ij,r"'=c*d-c^dv+- cHv^ - -l—cdv^~c'y, and so forth. 

 2 ^ 2 2.3 



1 z 



Hence the angle FGH is equal to (\cd — cy} — — [cV 



*' 2c + 2cd ^ X 



+ cdv + c^2/] ii + [ed - cVu + -L cdv- - cV] -^ - [c*d 

 X 2 X 



