8 



SIXTH REPORT. — 1836. 



Kater, he arrived at dimensions differing so much from those given 

 b}' that eminent author as to lead him to fear that he had made some 

 gross error in applying the differential calculus to the investigation. 

 A little consideration, however, convinced him that the fundamental 

 principle of Kater's calculation was erroneous. 



The erroneous principle virtually adopted by Captain Kater is, that 

 the centre of oscillation of the heavy metallic part of the pendulvun re- 

 tains constantly its relative place in the mass ; so that its distance from 

 the lowest part where it is supported by the pendulum rod is to be 

 taken as the length of metal whose expansions and con- g 

 tractions are to compensate those of the rod. Now it is 'p ^ 



almost obvious that the position of that centre in the mass 

 changes, on two accounts : first, the moment of inertia 

 of the mass which is the numerator of part of the value 

 of the length of the pendulum is changed by the chan- 

 ging of the dimensions of the several parts of the pendu- 

 lum by changes of temperature ; secondly, the distance 

 of the centre of gravity of the mass from the axis of sus- 

 pension changes also, and it enters as a denominator into 

 the same value. These combined causes produce a 

 change of great practical consequence in the position of 

 the centre of oscillation during alterations of temperature. 



Time permitted Mr. Stevelly to exemplify these re- 

 marks only in the pendulum composed of a deal rod sus- 

 pended by a steel spring, and a leaden tube. This pendu- 

 lum is perhaps the cheapest, simplest, and best that can 

 be made. 



Let the annexed figure represent a deal rod and leaden 

 tube pendulum ; S P = 2 inches of steel spring ; P D the 

 length of the deal rod to be calculated ; L D = 2 z = 

 length of leaden tube ; B a deal circular bracket, either turned and 

 fastened upon the end of the deal rod or made out of the same solid 

 piece of white deal wood with the rod, its use being to give a firm 

 support to the leaden tube. Let 2 r = the outside diameter of the 

 leaden tube, and 2 r' equal the diameter of the cylindric hole along 

 its axis which receives the deal rod ; let G be the centre of gravity, 

 and O the centre of oscillation. Let S G = \ and S O = (for a royal 

 seconds pendulum) 39" 13929 inches : denote this by /. 



It can then be easily shown by the formulae for centre of oscilla- 

 lion, that 



, 4 3 , 4 3 .„s 



¥ 



A X 



By applying the differential calculus to find the change of po- 

 sition of O for changes of temperature, we shall see that since / is 

 constant. 



d(/-A)= -dA = 



'2rdr+2r'dr' 2zdr / r"^ + r" ^ _^ ^\ ^^ 

 _ 4 "•" 3 V 4 3/ ■ 



A- 



