562 PROFESSOR CHRYSTAL ON LAGRANGE'S RULE. 



this process is no better than Lagrange's " Demonstrate Adumbrata," for the question 

 how far the equation (A) derived in this way from (E) is really equivalent to (E) is not 

 discussed at all. We now know that (A) is not equivalent to (E). Of this Jacobt 

 himself seems to have been in some measure aware, for he says : — " Extat interdum 

 ?equationis (E) solutio, quae neque ipsa Constantes arbitrarias implicat neque e solutione 

 Constantes arbitrarias implicante provenire potest valores iis tribuendo particulares. Quae 

 solutiones per methodum anticedentibus traditam ex sequatione (A) solutionibus elici 

 nequeunt, sed, si extant, absque omni integratione inveniuntur. De quibus solutionibus 

 singularibus hoc loco non agam." 



In Boole's Differential Equations (2nd ed. ( 1865, p. 332) a demonstration is given 

 in which the confusion between a conditional equation and an identity occurs twice over. 



Serret (Calcul Integral, 1868, p. 601) comes nearer the true demonstration. He 

 points out that we can express >\r(x,y,z) in the forms Xi( a W y )> Xt{ y > u > v )> ano ^ stows 

 that 



ox cy 



He then reasons that, since SxV^ anc ^ ^X^V are D Y hypothesis not identically null, the 

 solution -v^ = must make P = 0, Q = 0, E = 0, for " we cannot admit that S^i/S^ = and 

 &Xi/&y = can De satisfied as a consequence of -%i — or ^ 2 = 0, for the elimination of x or 

 of y would give a final equation between u and v, which implies contradiction." He 

 seems to forget that it may happen that Sp^/Sx and ^ 2 may have a common u-v- factor, 

 or otherwise be such that the two equations 8xi/^ x = 0> Xi = ^ are no ^ independent. This 

 is what actually does happen in general. See the examples given above (p. 557). 



Imschenetsky (Archiv der Mathematik und Physik, Th. 1., 1869, p. 295) reproduces 

 the demonstration given by Lagrange in the Calcul des Fonctions without alluding to 

 the fallacy it contains. 



The demonstration given by Graindorge (Memoire sur V Integration des Equations 

 aux Derivees Partielles, 1872, p. 12) contains the usual fallacy of confusion between an 

 identical and a conditional equation. The same remark applies to the proof given by 

 Mansion (Tlieorie des Equations aux Derivees Partielles du Premier Ordre, 1875, 

 p. 35), which is in form the same as that given by Forsyth. It is needless to criticise 

 the demonstration given by Jordan (Cours d 'Analyse, t. iii. p. 312), because, even if it 

 were clear and convincing so far as it goes, it does not pretend to be complete. 



This little piece of mathematical history shows us that human judgment even of the 

 best judges is fallible even in such a simple thing as mathematical reasoning. It teaches 

 us to distrust general demonstrations which are not built on the most obvious elementary 

 foundations. All such demonstrations should constantly be brought to the test of parti- 

 cular instances. The difficulty in proving Lagrange's Rule was no doubt insuperable 

 until the nature of singular solutions had been determined, and the cases separated in 

 which it was not true ; but this was no excuse for the bad logic of the pretended 

 demonstrations. 



