406 



the Academy the day this second paper of Clairaut's 

 was read erroneously, for Fontaine had shown it in 

 November, 1738 ; and had said that it was then new at 

 Paris, and was sent from thence to Euler and Bernouilli. 

 The probability is, that Clairaut had discovered it in- 

 dependent of Fontaine, as Euler certainly had done ; 

 and both of them handled it much more successfully 

 than Fontaine. D'Alembert, in his demonstrations, 

 1769, of the theorems on the integral calculus, given 

 by him without any demonstration in the volume for 

 1767, and in the scholium to the twenty-first theorem, 

 affirms distinctly that he had communicated to Clairaut 

 a portion of the demonstration, forming a corollary to 

 the proposition, and from which he says that Clairaut 

 derived his equation of condition to differentials involv- 

 ing three variables. It is possible ; but as this never 

 was mentioned in Clairaut's lifetime, although there 

 existed a sharp controversy between these two great 

 men on other matters, and especially as the equation of 

 conditions respecting two variables might very easily 

 have led to the train of reasoning by which this exten- 

 sion of the criterion was found out, the probability is, 

 that Clairaut's discovery was in all respects his own. 



The extreme importance of this criterion to the 

 method of partial differences, only invented, or at least 

 applied, some years later, is obvious. Take a simple 

 case in a differential equation of the first order, 



d z = (2 a x y y*) dx + (ax 2 3 a; y 2 ) dy 



where M = 2 a # y y 3 , N = a # 2 3 # ?/ 



For the criterion = 2 a x 3 y 2 

 dy 



gves us 



dy dx 



