326 Mr. Ivory's Remarks on M. Poisson's Memoir: 



The difficulty is now overcome ; for there is no doubt that the 

 integral is equal to 47r. The integration may be performed 

 as usual ; or it may be accomplished by the peculiar process 

 of M. Poisson. The argument cannot be affected by different 

 algebraic operations that lead to the same result. But we 

 may fairly hesitate to admit the gratuitous supposition of 

 making if constant. In order to examine this point, put 

 y = j/ + (y — j/) * : then 



Y — y / »(X -ar-)ds 1 r (1 -!>.•') {y -y)ds 



A^j /» "^ 4 ^ y p 



If we neglect the term newly introduced, what remains is 

 M. Poisson's demonstration. Whether we admit or reject 

 his conclusion, will therefore depend upon the evidence we 

 have that the term omitted is evanescent. Now put this term 

 equal to zero ; separate it into the two parts of which it con- 

 sists ; and substitute the known value of the integral multi- 

 plying the constant quantity y : then. 



But this Is neither more nor less than the original formula to 

 be demonstrated, if we substitute y for X. It appears, there- 

 fore, that the very property to be pi'oved is involved in the 

 omitted term ; or, which is the same thing, in the assumption 

 made by M. Poisson, that y' is constant. The boasted de- 

 monstration published in 1823f, which was to dissipate all 

 doubts and objections, is merely a petitio prhicipii. I am 

 induced to make such observations, because I am concerned 

 to show that the objections I have made are not frivolous, but 

 such as it would be a reproach to any one to overlook them 

 in the profest examination of a difficult question. 



But, in his new Memoir, M. Poisson endeavours to correct 

 his former demonstration, by considering the term which must 

 be taken into account, in order to confer rigour and accuracy 

 upon the reasoning:]:. To use a homely phrase, he makes no 

 bones of it. He resorts to his former assumption, and inte- 

 grates on the supposition that y' — ?/, or ^ as he denotes it, is 

 an infinitely small constant quantity. By this means the term 

 in question comes out infinitely small, or zero: and this is all 

 which is thought necessary for settling the point in dispute. — 

 Will this pass for demonstration ? It is a mere assertion. It 

 is one of those curt and imperative attempts at pi'oof, of which 

 too many occur in the modern mathematics, which are none 



* Phil. Mag. for Jan. 1826, pp. 36, 37. 



\ Journal de r Ecole Poly technique, IJ)" cahiei", p. 145. 



X Conn, des Terns 1820, pp. 332, 333. 



of 



