356 Astronomical and Nautical Collections. 



2 2.3 ^ ^J \^ ^^7 ^ 5 



^^ A)...). And the sum of the angles FAG, FGH, that is GFH == 



zz 



\ (cd — - (c^d -f cc/r) — + (c'd — c~dv + J- cdv') 



i^ - (c^c^ + cWt; - £!^ + i^") r l.3.5fto 1 ^^T 

 0.'^ ^ 2 2.6 y L a;' 5 z* ■• 



Where the angle SAD, or the zenith distance, is small, the angle 

 GFH may be conveniently found by this series ; but when it be- 

 comes greater, we must find the angle FGH by the former series. 



Another series may also be obtained for the angle FGH, by the 

 seventh proposition, [containing the author's well known theorem]. If 



Qbe the fluent of ~ ~^M or of xy ; it follows from that proposition, 



X 



thatwhlle-rbecomes or ±f,Q will become Q±—u + r" ib 



X 2i' 2.3i' 



v^ + . . .; that is, supposing x to flow uniformly. If, therefore, 



we take for x its value in any point I, and x — d be its value in A, 



and X + V in u, the value of the fluent in the point A will be 



Q+ —-V + ... and in the point a it will be Q — —u +• .which 



X X 



being deducted from the former, the remainder will be the part 

 corresponding to the line Aa; and if SB ::^ " " , the angle FGH 



will be _J_ f^v +:^ + J^ +^ + ...). In this case, 

 c + cd \x 2.3x^ 



if we call x — \, we have z = — , and y s Zj-^ : and Q' 



z cz 



being^,,Q-=_i^(i^-iiYQJ_^(:^-^ 

 czz \ c z J cz \cc cz 



,2tt , >, 1 + 5x' \ 



Scholium. The radius of curvation of this curve is 

 (2 + 2^)0 xSB cub. ^^.^^ . ^ ^^^ ^ .^ (2+2d)cxSA, 



2/xSQxSAq ^ ri+SD 



