826 Biographical Account of [Mat, 



culties considered as insurmountable, and to rectify or complete 

 theories remaining imperfect, appears to liave always directed M. 

 Lagrange in the clioice of his subjects. 



D'Alembert had considered it as impossible to subject to calcula- 

 tion the motions of a fluid inclosed in a vessel, unless iiiis vessel had 

 a certain figure. Lagrange demonstrates the contrary; txcept jn 

 the case when the fluid divides itself into difterent masses. But 

 even then we may determine the places wlierc the fluid divides it- 

 self into difterent portions, and determine the motion of each as 

 if it were alone. 



D'-llembert had though: that in a fluid mass, such as the earth 

 inay have been at its origin, it was not necessary for the different 

 beds to be on a level, Lagn.; ge shows that the equations of 

 D'Alembert are themselves equations of beds on a level. 



In combating D'Alembert with all the delicacy due to a mathe- 

 matician of his rank, he often employs very beautiful theorems, for 

 wln'ch he was indebted to his adversary. D'Alembert on his side 

 added to the researches of Lagrange. " Your problem appeared to 

 me so beautiful," says he in a letter to Lagrange, *' that I have 

 sought for another solution of it. I have found a simpler method 

 of arriving at your elegant formula." These examples, which it 

 would be easy to multiply, prove with what politeness these cele- 

 brated rivals corresponded, who, opposing each other without in- 

 termission, whether conquerors or conquered, constantly found ia 

 tlieir discussion reasons for esteeming each other moie, and fur- 

 nished to their antagonist occasions which mlaht lead them to new 

 triumphs. 



The Academy of Sciences of Paris had proposed, as the subject 

 of H prize, the tlieory of the libration of the moon. That is to 

 say, t!'ey demanded the cause w hy the moon, in revolving round 

 the earth, always turns the same face to it, some variations ex- 

 cepted, observed by astronomers, and of which Cassini had fii-st 

 explained the mechanism. The point was to calculate all the plie- 

 nomena, and to deduce them from the principle of universal gravi- 

 tation. Such a suljject was an appeal to the genius of Lagrange, 

 an opportunity furnished to apply his analytical principles and dis- 

 coveries. The attempt of D'Alembert was not disappointed. The 

 memoir of Lagrange is one of his finest pieces. We see in it the 

 first developement of his ideas, and the germ of his Metanique 

 Analytique. D'Alembert wrote to him : *' 1 have read witii as 

 much jdeasure as advantage your excellent paper on the Libration, 

 so worthy of the prize which it obtained." 



This success encouraged the Academy to propose, as a prize, the 

 theory of the satellites of Jupiter. Euler, Clairaut, and D'xAlem- 

 bert had employed thcmseives about the problem of three bodies 

 on occasion of the movements of the moon. Bailly then applied 

 the theory of Clairaut to the problem of the satellites, and it had 

 led him to very interesting results. But this theory was insuflicient. 

 Tiie earth has only one moon while Jupiter has four, which ought 



