264 On Mr. Herapath's Second Attack on Lagrange's Method. 



which is the complete integral in its simplest form when the 

 roots are equal. If we suppose X = 0, then, 



y — e rx (h + c <r). 

 Thus, when we follow a proper method, it appears that the 

 complete integral comprehends all the subordinate cases; 

 which proves the incorrectness of Mr. Herapath's views. 

 , Nothing more was intended by the notice inserted in this 

 Journal for February last, p. 96, but to correct Mr. Herapath's 

 mistake about Lagrange's method. It may now be proper to 

 inform him that there is nothing new in his papers. Euler 

 has integrated the equation of the third order in his Calculus 

 Integrates, torn. ii. sect. ii. cap. 3. prob. 149; and at the be- 

 ginning of the scholium to the problem, he observes : ". In 

 genere autem, nulla integralium reductione adhibita, integrale 

 nostrae equationis ita exprimi potest ;" and he then sets down 

 the very same expression of y which is contained in Mr. He- 

 rapath's equation 19. Euler's words contain the exact de- 

 scription of Mr. Herapath's method, which finds the integral 

 by successive integrations without attempting to reduce it to 

 the form best adapted for use. In prob. 151. of the same chap- 

 ter, Euler exhibits the unreduced integral of any order inde- 

 finitely, being the very same with Mr. Herapath's equation 17, 

 and therefore containing the sum and substance of that gentle- 

 man's doctrine. Lastly, in prob. 152. cor. 4. it is observed 

 that this form of the fluent is free from the difficulty about 

 the equal roots, on account of the absence of zero divisors. 

 These accumulated proofs show that there is nothing new in 

 Mr. Herapath's papers, except the extraordinary mistakes they 

 contain. 



The history of this problem ; the real difficulties attending 

 it; the slips that were at first made by geometers of the highest 

 eminence about the case of equal roots ; the correction of these 

 slips, and the artifices by which the solution has been com- 

 pleted ; — all these are points about which it is fair to presume 

 that Mr. Herapath is at present not well informed. 



Some apology is due to the readers of this Journal, for the 

 length of these remarks ; but it seemed proper to explain the 

 matter pretty fully ; for, unless the writer of this article has 

 been misinformed, Mr. Herapath's solution of this problem 

 has, not much to the credit of British science, been blazed 

 abroad as a great discovery. I now withdraw from any further 

 intermeddling in this trite subject *. n 



XLIII. On 



* It is probable that his Postscript relates to this frivolous point, viz. 

 whether the arbitrary constant in p is to be written with the same deno- 

 minator 



