a general Formula for the Length of the Seconds Pendulum.2^3 



sists of 39 English inches and a small variable part, it will free 

 the arithmetical operations from the embarrassment of large 

 numbers, if we put /= 39 + 8, V — 39 + 8',&c. L = 39 + A : 

 then 



A + /sin 2 A — 8 = e 



A + /sin 3 A'-8'=^ 



A + /sin 2 A"-8"= e" (Aj 



&c. 

 In these equations the errors e, e', &c. are functions of A and 

 f; and it is evident that, 



de de' Q 



dA ' dA 



— e - = sin 2 A, -jT- = sin 2 A', &c. 



df df 



Take the squares of both sides of every one of the equations 

 (A), then (A + /sin 2 A - 8) 2 = e 2 



(A + /sin 2 A'-8') 2 = d°~ 



&c. 

 Differentiate these equations, making A only vary, and sub- 

 stitute the values of the differential coefficients, — , -r-,&c. : 



' d& d A 



then, A + /sin 2 a — 8 = e 



A + /sin 2 A'- 8'= d 



&c. 

 Further, n denoting the number of the experiments, put 



g _|_y + r 4- & c. 



A : , 



re ■ 



•p, sin 2 X. + sin a X 7 -f- sin a A." + &c. 



e + e 1 +e' + &c. 



°" ~ r~^ s 



re 



then, by adding the foregoing equations, we get, 



A + B/ - A = cr. 

 Again, differentiate the same equations as before, making f 



only vary, and substitute the values of —rz, —■, &c: then 



A sin 2 A + /sin 4 A — 8 sin 2 A = esin 2 A 

 A sin 2 A' + /sin 4 A'- 8' sin 2 A' .= d sin 2 A' 

 &c. 



Put now, 



n _ sin* A. + sin* A/ + sin* A." + &c. 



re 

 -p. Ssin^A. + ysing*/ + g"sin3A." + &c. 



re 

 (e — «■) sin 3 A. -\- [e '— <r) sin a A.' -)- &c. 



ihen, 



