as deducedfrom Experiments made with the Pendulum. 99 



minimum. Now, in this it is evidently assumed that the con- 

 ditions of the problem can be exactly fulfilled in no other case 

 except the contemporaneous evanescence of all the errors. 

 But this supposition is too limited ; for the conditions will be 

 exactly fulfilled when the errors are all equal to the same quan- 

 tity s, although they be not all evanescent. What obstructs 

 the perfect solution of the problem is the circumstance that 

 x^ l \ x^\ a? (3) , &c. are not all equal to zero; and we shall ap- 

 proach the nearest to a perfect solution when we assign to these 

 quantities the least values that the case will admit of. 



Applying the usual rule to the foregoing equations of con- 

 dition, we get this equation for finding/ viz. 

 = 3-76723/— 0-78207, 

 the coefficient of f being the sum of the squares of all the co- 

 efficients of /in the several equations, and the other number, 

 the sum of the products of the two numbers in each equation. 

 Hence we get f = 0*2076 ; and the formula for /, the length 

 of the pendulum at any latitude A, is 



I — 39*01175 + a + 0-2076 sin 2 A. 

 We may determine the arbitrary quantity s in different ways. 

 If, in a set of experiments, any one is entitled to much greater 

 confidence than the rest, s may be determined so as to make 

 the error of that experiment equal to zero : and, if there be 

 no reason for preferring one experiment to another, e may be 

 determined so as to make the sum of all the errors evanescent. 

 The following table contains the pendulums calculated by the 

 formula on the supposition that e = 0, together with the ex- 

 cesses of the calculated above the observed quantities. 





Calculated 



Excess of 



Station?. 



Pendulums. 



Calculation. 





inches. 





Maranham . . . 



39-01214 



•ooooo 



Sierra Leone . . 



39-01627 



— -00370 



Trinidad .... 



39-01884 



•ooooo 





39-02223 



— -00202 





39-03143 



- -00367 



New York .... 



39-10006 



— -00162 





39-13896 



— -00015 



Drontheim . . . 



39-17780 



+ -00324 



Hammerfest . . 



39-19659 



+ -00140 



Greenland .... 



39-20459 



+ -00124 



Spitzbergen . . . 



39-21288 



+ '00181 



As the error is very small at London, there seems to be no 

 N 2 good 



