Method of integrating Linear with Constant Coefficients, 211 



to be all equal, the three equations of condition, p. 23, which 

 are asserted, p. 96, to give the " complete integral " of 



S- + A ^ + B^ + Cj , = X (a) 



by Lagrange's method will become 



r 2 e rx {dp + dp, + dp,} = X 

 dp + dp, + dp 2 =0 

 dp + dp x -f dpi = 0, 



giving obviously X =0. That is, the equations of condition, 

 which are said to furnish the " complete integral " of (a), and 

 which therefore ought to give all cases of the integral without 

 limiting the value of X, fix in the above instance the value of 

 this indefinite function to zero. 



Any candid mind would not need a more decided proof of 

 absurdity in the principles of Lagrange's method than the 

 above ; but as some may think it dealing a little too summarily 

 with a process which has obtained so much celebrity, and may 

 prefer an instance of failure in result to a theoretical exposi- 

 tion of absurdity in principle, I submit the following, taken 

 not from any investigation of mine, but copied from a pro- 

 fessed advocate of Lagrange in the last Number of this vo- 

 lume. We are informed, p. 96, that the " complete integral" 

 of (a) is 



« y _ e rx {e+fXe- rs d x} 

 " r—r. . r—r 



1 2 



g r ' x {c l +/X«- r '*d*} b) 



r-r 1 .r 1 -r a 



e r * X {c 2 +fXe- r2X d*} .„ 



Now this being the " complete integral," must give the 

 value of y whatever values be assigned to r, r„ r, and X. 

 Putting for simplicity X = 0, and then supposing r = r, the 

 above equation becomes 



i r 2 x 



rx c—c x ce 



f " Ox(r-r c ) ^ (r-«jR 



that is, the value of y is infinite unless the arbitrary constants 

 c, c, are equal, or certain functions of r, r x . But c, c x being 

 in the strict sense of the word " arbitrary constants," are not 

 necessarily functions of x or r,r,; and therefore the infinite 

 value of y remains. We however know from other principles 

 that y may be finite. 



E2 Lest 



