22 Mr. Herapath on the Integration 



posed by Lagrange, which has been received and relied on as 

 correct by the scientific world for nearly half a century. That 

 we may treat this method with perfect justice and fairness, 

 we propose to work out as directed the identical example 

 with which it is in all the works of the present day illustrated, 

 and then show its absurdity by direct differentiation. The 

 equation usually taken is one of the third order, 



d 3 y + Ad 2 y + Bdy + Qy = X. (20) 



Suppose e rx , e r * x , e* 2 *, or y l9 y 2f y 3 be the three partial solu- 

 tions when X = ; then if we take 



y = pe rx +p 1 e TlX + p 2 e r * x or = py, + p,y 2 +p 2 y 3 



p, p l , p 2 being instead of arbitrary constants arbitrary func- 

 tions of x, these arbitrary functions may, it is conceived, be 

 subjected to certain restrictions, and yet satisfy the conditions 

 of (20). For instance in the value of dy in 



dy=pdy, + p x dy 2 + p 2 dy 3 +y t dp + y 2 dp l -ty 3 dp 2 



the first three terms are the same as would result if p, p x , p 29 

 were constant coefficients ; and Lagrange therefore assumes 

 that he may put 



y, dp + y 2 dp, + y 3 dp 2 = 

 as one condition for determining the functions p 9 p x , p 29 which 

 leaves dy = p dy, + p, dy 2 + p 2 dy 3 



This being a second time differentiated becomes 



d 2 y = pd 2 y t +p l d*y i + p 2 d% + dy x dp + dy 2 dp, + dy 3 dpi 



which for similar reasons Lagrange again decomposes into 



d 2 y = pd z y, + p x d*y 2 + p 2 d*y 39 



and dy, dp + dy z dp t + dy 3 dp 2 = 



Differentiating once more we have 



d 3 y=pd?y v + p v d 3 y 2 + p 2 d 3 y 3 + d 2 y, dp + d 2 y 2 dp x + d%dp 2 



Substituting the several values of y 9 dy 9 d 2 y 9 d 3 y for these 

 quantities in the proposed equation there results 



p {d 3 y f + k#y x + Bdy, + Cy x } 



+ Pi Kj/2 + Ad*y 2 + Bdy 2 + Cy,} 



+ p, {d 3 y 3 + Ad 2 y 3 + Bdy 3 + Cy 3 } 



+ cPy,dp + (PyAPi + d*y 3 dp 2 = X 



But because the parts multiplied by p, p x , p 2 are assumed 



to 



