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XV. On Mr. Herapath's Method/or the Integration of Linear 

 Equations with constant Coefficients, 



T^HE integration of linear equations with constant coeffi- 

 ■*• cients engaged the attention of analysts very early in the 

 progress of the Integral Calculus. The subject has been treated 

 of so often and so ably, that we can hardly expect any addi- 

 tion to our knowledge of it either new in principle, or very 

 interesting in the details. It would not therefore have been 

 necessary to notice particularly the communications of Mr. 

 Herapath published in the last two Numbers of this Journal, 

 if he had not fallen into an inadvertence respecting Lagrange's 

 investigation, of which it seems proper to advise the public. 



In pp. 22, 23, of the last Number of this Journal, Mr. He- 

 rapath applies Lagrange's process to the equation, 



dx3 



+ A S + B if + c 3' = x. 



The calculation is given at length in Lacroix's Treatise on the 

 'Integral Calculus, 2nd edit. vol. ii. pp. 326, 327. If r, r v r % 

 stand for the roots of the equation 



I* + Ar 2 + Br + C = 0, 

 the complete integral is as follows, viz. 



e rx (c+f*e- rx dx) 



y a- 1 . 



Y r—r t . r—r 2 



e r '*( C ,+/X g -^rf* >) 

 r l — r.r l —r.z 



r.2 — r.r2 — r t 



Of these three parts Mr. Herapath has calculated only the 

 first, which he says must separately satisfy the proposed equa- 

 tion. Now in this assertion, of which no proof is given, lies 

 his mistake. If every part had contained only one root, it 

 might have been rightly inferred that it must separately satisfy 

 the differential equation ; but, as the case stands, there is no 

 ground for the assertion. If Mr. Herapath will substitute the 

 two remaining parts of the integral in the differential equation, 

 in like manner as he has substituted the first part, he will find, 

 on collecting the three results, that there is no failure in La- 

 grange's method. 



«/3. 



XVI. On 



