258 Mr. J. Rose-Innes on the Theory of 



Since 



we must also have 



This equation is exact; but if [ -7— J and JK ^— are small 



quantities, and if we are content to neglect squares of small 

 quantities, we may put approximately 



that 



\dpJt Op f \vj 



f(v) 

 The expression '-^ — +1; is a function of v only, and as we 



f\ v ) 



know independently 2 i-t— ) + JK ^— is a function of t only 

 v J \<-tp It dp 



, 1 /(«> 

 we must nave -^ — r 



differential equation 



we must have AM; + *' a constant say B. Then from the 



^ + .-B, 



we obtain by integration 



R 



whei*e E, is a constant. The constant It has been introduced 

 by an integration, and is therefore arbitrary as far as the 

 differential equation is concerned ; it is fixed, however,, by 

 the consideration that the interval of temperature between 

 the freezing-point and the boiling-point has an arbitrarily 

 chosen value. Since 



X 



