72 Chemical and Physical Notes 



amount present at the beginning of the 1st second which is 

 represented by unity. Therefore, referred to the amount 

 present at the zero of reckoning as unity, the amount re- 

 maining at the end of the 2nd second should be (^) 3 . 

 Similarly, during the 3rd second the thermometer loses 

 Y^ of the heat which it had at the beginning of that second, 

 and the amount remaining at the end of the 3rd second will 

 obviously be $& of ( T 9 ^) 2 , or (f^,) 3 . Similarly, at the end of 

 the 4th, 5th, 6th, or th second, the heat remaining will be 

 (f$>) 4 > (f&) 5 > (f&) 6 > .-., (W of the original amount present at 

 the zero of reckoning represented as unity. It is obvious that 

 if the law holds good, by giving the suitable value to the 

 index n we can at once calculate the proportion of heat which 

 will remain after any number of seconds of cooling ; and it is 

 also obvious that so long as the temperature of the medium 

 remains constant, the thermometer can never exactly reach 

 that temperature, although the difference of the temperatures 

 may be made as small as we like by making the duration of 

 cooling sufficiently long. 



It has been said that by giving n the suitable value, we 

 can at once find the heat remaining after the lapse of any 

 time. But the computation of high powers of numbers by 

 ordinary arithmetic is very laborious. If, instead of simple 

 arithmetic, we use logarithmic arithmetic, the computation of 

 a high power is as easy and expeditious as the computation of 

 a low one. In the case which we have imagined the constant 

 fraction is -$fa. Its logarithm is log 99 - log 100, that is 



i'9956352 - 2 = 0-9956352 - i, 



which is usually written 1*99563 5 2. 



This is the logarithm of the first power of T 9 9 ^, which is 

 the heat remaining at the end of the ist second of cooling. 

 If we multiply 1*9956352 by 2 we have the logarithm of the 

 square of ffo, and if we multiply it by 3, 4, ..., n we have the 

 logarithms of the 3rd, 4th, ..., th power of T %%, that is, of the 

 heat remaining at the end of the 3rd, 4th, ..., th second. 

 These logarithms differ from each other by the same amount. 

 Therefore we have the following rule : If the initial excess of 



