THE AXES OF A QUADRATIC VECTOR. 351 



14. Application to Consecutive Chemical Processes. 



As another illustration of the utility of the coefficients An etc., 

 it may be noted that an important class of chemical processes, namely, 

 the type known as "consecutive," leads to a pair of differential equa- 

 tions of the form 



at dt 



where Pi and P2 are quadratic polynomials, (not in general homogene- 

 ous), in the dependent variables .ti, .T2, but are not functions of t. 



These equations cannot in general be solved by quadratures, but 

 by a proper choice of the conditions of the process they may often be 

 made integrable in this way, and the labor of solving by series avoided. 

 We have merely to render the polynomials Pi and P2 homogeneous 

 by introducing a third variable xz and write down the nine A's for the 

 quadratic vector 



/3lPl + i32P2 



which will always have at least one set of coplanar axes since it is a 

 binomial. If then we can so choose our conditions that, according to 

 the tests of Art. 9, there are three other sets of coplanar axes, the 

 equations can be solved by quadratures. 



