44 DYNAMICAL PRINCIPLE. 



wards ascertaining that function by attending to the other 

 conditions of the question. This method is called that of 

 partial differences. Lacroix justly observes that it would be 

 more correct to say partial differentials ; and a necessary part of 

 it consisted of the equations of conditions, which other geometri- 

 cians unfolded more fully than the inventor of the calculus 

 himself; that is to say, statements of the relation which must 

 subsist between the variables or rather the differentials of 

 these variables, in order that there may be a possibility of 

 finding the integral by the method of partial differences. It 

 appears that Fontaine, a geometrician of the greatest genius, 

 gave the earliest intimation on this important subject ; for 

 the function of one or both variables which is multiplied by 

 d x being called M, and that function of one or both which is 

 multiplied by d y being called N, the canon or criterion of 

 integrability is that 



dy dx 



and we certainly find this clearly given in a paper of Fon- 

 taine's read before the Academy, 19th November, 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 Bernoulli. The 

 probability is, that Clairaut had discovered it independent of 

 Fontaine, as Euler certainly had done ; and both of them 

 handled it much more successfully than Fontaine. D'Alem- 

 bert, 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 



