TRANSACTIONS OF THB SECTIONS. 



Hence 



(r^+r''^ z"A ^rdr-\-2r'dr' , 2 

 x^ /= K 



( 2rdr+ ^r'dr' 2zdz 

 4 "'■ 3 \ 

 7~r~r. — ^ — I 



Now if we denote hj dm the change of length which the unit 

 length of the metal of which the tube is formed, suffers from 

 the change of temperature to which the pendulum has been sub- 

 jected, then dr = rdm: dr' =^ r' dm: and dz = z dm; and substi- 

 tuting these, we have 



/<2 ^_ ^.'2 2 «« 





<!('-*)=( ..4-.. .2 ''"'' 



ITie height of the point O, above the bracket which supports it 

 is, = 2 — (/ — X) . Hence the change of place of O, upwards or 

 downwards in relation to the bracket, is the differential of this, and 

 is therefore equal to 



/r^-l-r'a 2z^\ 





dm 



4~ ~ 



Now the coefficient of dm in this expression is manifestly the length 

 of the metallic part of the pendulum, whose changes for tempera- 

 ture are to compensate the changes of the suspending rod ; whereas 

 the length of that metal, according to Kater, is the height of o above 

 the bracket, which is 



— -^- — -H — 

 ., s 4 3 



In other words, this is Kater's coefficient for dm; and since 



r'2 2 z"-\ r« -f- r'^ 



/r"--|-r'2 2z"-\ r* -f- r'2 



4 3 r^ + r'- z^ /v 



^- 4 T 



Consequently, for any given z, Kater's coefficient will be greater 

 than the true coefficient ; therefore, it hence appears that he will 



