Astronomical and Nautical Collections. 359 



F' (x,u) a function of- a; and u capable of limitation. For the 

 same reason, keeping always in view the most general form of 

 the functions of x and u, we must have 



F' {x,u) = F' X + V F" {x,u), 

 F" (x,u) = ¥" X + V" F'" {x,u). 

 F" (x,u) = F'" X + V" F"" (x,u), &c. 

 V; V", 4-c., being functions of m without limit in diminution, 

 and F" (x,m) ; F'" (^x,u) ; &c., functions of x and u capable of 

 limitation. Now the first of these conditions cannot be ex- 

 pressed in a more general manner, than by making 



V ri M (p w. 



V = M (p' u. 



V" = u tp" u, &c. 

 expressions in which (pu; (p'u; (p"u, &c., represent functions of 

 u capable of limits, or constant quantities : and the substitution 

 of these values, in the former equations, reduce them to 



F (ot+m) = F X + u tp u ¥' {x,u). 



F' {x,u) ~ F X + u(p' u F" {x,u). 



F" {x,u) = F" X + u (p" u I"" (x,m), &c. 

 and the substitution of each of these in the others gives us 

 finally, 



F(x+m) = Fx+u(pu F'x+u^(pu(p'u F"x + u^(pu<ii'u(p"uF"'x+ &c. 

 or, if we write x for v, and u for x, which in no way changes 

 the function F(x + u). 



F(_u + x) = Fu + x(px F'u+x-tpxtp'x F"u+x^(px(i)'x(p"x F"'it + &c. 

 If we here observe, that u has no limit of diminution, and 

 if we denote by FO, F'O, F"0, &c., the limits of Fm, or the values 

 to which these funttions are reduced by the substitution of a 

 zero foi the symbol denoting its root, we shall obtain ultimately 

 from this formula the following equation. 

 Fx = FO + x(px F'O + x'lpx (p'x F"0 + x^tpx (p'x (p"x F"'0 + &c. 

 and in this most general expression consists the solution of the 

 problem proposed. 



In the subsequent sections the author proceeds to introduce 

 more particular values, for such of the quantities as here remain 

 indeterminate : but you will be able to judge of the method that 

 he employs by the first section, of which I have given you a 



