414 Mr. G. Boole 07i the Sohttion of a 



in which TTm, 7r„, and p satisfy the conditions 



ir,npli=pTr,n+iU, 7r„pu=p7r„+iU 



7r,„ 7r„ ti = 7r„ 5r,„ U + a{il — m)pU 



(Mr. Bronwin has changed p into \ and y into p). Of the 

 above equation I assigned two complete forms of sohition, anil 

 I showed that Laplace's equation might be treated as a very 

 particular example of the more general problem thus intro- 

 duced. But at this stage of the inquiry, a transformation 

 suggested itself to me by which Laplace's equation was made 

 to satisfy the condition to which 1 had previously failed to 

 subject it. This led to a resumption of the prior method, and 

 to the complete accomplishment of all that I at first contem- 

 plated (Cambridge Mathematical Journal, New Series, vol. i. 

 p. 10). It led also to the abandonment of any further re- 

 searches on the equation (2.). What had been accomplished 

 was, however, published in the same journal, Jan. 184'7. It 

 is this unfinished episode in the investigation, if I may be 

 allowed the expression, which Mr. Bronwin has undertaken 

 to complete ; and it is to the correction of some misapprehen- 

 sions which I venture to think that he has formed as to the 

 principles upon which such investigations are to be conducted, 

 that I am here desirous of contributing. 



Mr. Bronwin rightly apprehends that the solution of the 

 general equation 



TTm-rtnU + qpU^'K 



would be 



-1^-1 _-i ^-1 X; . (3.) 



but he conceives that I have erred in rejecting the inverse 

 factors 



in the particular case of X = 0. He asserts that it is always 

 necessary to retain such inverse factors, and to determine the 

 values of the arbitrary constants by final substitution in the 

 original equation ; and he adduces, by way of illustration, the 

 equation 



There are here involved, if I am not deceived, both a false 

 analogy and an erroneous principle. First, the equation 

 adduced is not analogous to the one which was under consi- 

 deration, and it is not solved by an analogous process. The 

 relative position of the direct and the inverse factors of opera- 

 tion in the one case, is the reverse of what it is in the other. 



