ELIMINATION OF CONSTANTS. 267 



dx 



It will be found on trial, that if we take any one of the 

 differential equations of the first order, and differentiate twice, 

 we shall obtain the same result if we eliminate the two 

 constants involved in these three equations, as we have 

 already found in equation (5) of Art. 241. Also, if we 

 take any one of the differential equations of the second order, 

 differentiate once, and eliminate the constant involved in 

 these two equations, we shall still arrive at the equation (5) 

 of Art. 241. 



244. The process by which, as in the preceding Article, 

 we may deduce differential equations by differentiation and 

 elimination of constants, has not in itself much interest or 

 value. But the method of passing from the differential 

 equations to the primitive equation from which they were 

 deduced, forms a most important branch of mathematics. In 

 fact* all investigations in physical science lead to differential 

 equations, which must be solved before we can be said to 

 understand the subject we are considering. We do not 

 enter here on the solution of differential equations, but it 

 is usual, in treatises on the Differential Calculus to devote 

 some space to the consideration of the formation of such 

 equations by elimination, as this process throws light on the 

 methods to be adopted for their solution. 



