DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 5 



of a, 6, /, g, h, the preceding inference is not justified. Manifestly one such condition 



will be 



3F 3F _3F 3F 8F . 3F 



but this is, of course, only one among a number of equations. 



The method, practically contained in CAUCHY'S theorem, leads to a result only in 

 an infinite form ; moreover, it takes no account of the alternative when conditions are 

 not satisfied. We proceed, accordingly, to give another method, suggested by the 

 corresponding investigation (due to DARBOUX) for equations of the second order in 

 two variables. 



2. The principle underlying DARBOUX'S method is, in effect, similar to that which 

 underlies AMPERE'S ; but as DARBOUX'S method most easily admits of application to 

 equations in which the derivatives of highest order occur linearly, it is customary to 

 form derivatives of a given equation, in order to secure that the equations discussed 

 shall possess this property. If however the equation be already in a linear form, a 

 mere generalisation of AMPERE'S method can first be tried, for the number of 

 subsidiary equations is considerably smaller than in the other method, and conse- 

 quently the integrations (if they can be performed) are correspondingly easier.* It 

 will be sufficient for the present purpose to consider a particular example, say 



When we substitute for b, f, g, h, we find 



dm , , , . dn , dn dm . , N 



^+(l-ff-l)^ + S - S + (fl i -M + fir-Jp)o-0; 



the equation now must not determine the value of' c, and so we must have 



dm . -,\dn dn dm 



_ + (p _ g .. !)_ + --- = 



and there is also the identical condition 



dm dn dl dn 



_ f y\ _ ___ ^ _ I *V _ - . - I I 



da: " dy dy T-dy. 



From the first of these, it follows that either 



p g = 0, 

 or 



* A discussion of the subsidiary system, and of the relation of its integral to the solution of the 

 original equation, will be found later, in 27-29. 



