D'ALEMBERT. 417 



of the greatest genius, gave the earliest intimation on 

 this important subject; for the function of one or both 

 variables which is multiplied b j d x being called M, and 

 that function of one or both which is multiplied \>y d y 

 being called N, the canon or criterion of integrability is 

 that d M _ d N 



d y " d x 



and we certainly find this clearly given in a paper of 

 Fontaine's read before the Academy, 19th Nov., 1738. 

 It is the third theorem of that paper. Clairaut laid down 

 the same rule in a Memoir which he presented in 1739; 

 but he admits in that Memoir his having seen Fontaine's 

 paper. He expounds the subject more largely in his far 

 fuller and far abler paper of 1740; and there he says 

 that Fontaine showed his theorem to 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 independent of Fontaine, as Euler cer- 

 tainly 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 corol- 

 lary to the proposition, and from which he says that 

 Clairaut derived his equation of condition to differentials 

 involving three variables. It is possible; but as this 

 never was mentioned in Clairaut's lifetime, although 

 there existed a sharp controversy between these two 



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