376 HISTORY OF MECHANICS. 



1748, Euler not only assented to the generalization of D'Alembert, 

 but held that it was not necessary that the curves so introduced should 

 be defined by any algebraical condition whatever. From this extreme 

 indeterminateness D'Alembert dissented ; while Daniel Bernoulli, 

 trusting more to physical and less to analytical reasonings, maintained 

 that both these generalizations were inapplicable in fact, and that the 

 solution was really restricted, as had at first been supposed, to the 

 form of the trochoid, and to other forms derivable from that. He 

 introduced, in such problems, the "Law of Coexistent Vibrations," 

 which is of eminent use in enabling us to conceive the results of com- 

 plex mechanical conditions, and the real import of many analytical 

 expressions. In the mean time, the wonderful analytical genius of 

 Lagrange had applied itself to this problem. He had formed the 

 Academy of Turin, in conjunction with his friends Saluces and Cigna ; 

 and the first memoir in their Transactions was one by him on this 

 subject : in this and in subsequent writings he has established, to the 

 satisfaction of the mathematical world, that the functions introduced in 

 such cases are not necessarily continuous, but are arbitrary to the same 

 degree that the motion is so practically ; though capable of expression 

 by a series of circular functions. This controversy, concerning the 

 degree of lawlessness with which the conditions of the solution may 

 be assumed, is of consequence, not only with resj)ect to vibrating 

 strings, but also with respect to many problems, belonging to a branch 

 of Mechanics which we now have to mention, the Doctrine of Flui Is 



11. Equilibrium of Fluids. Figure of the Earth. Tides. — The 

 application of the general doctrines of Mechanics to fluids was a 

 natural and inevitable step, when the principles of the science had 

 been generalized. It was easily seen that a fluid is, for this purpose, 

 nothing more than a body of which the parts are movable amongst 

 each other with entire facility ; and that the mathematician must trace 

 the consequences of this condition upon his equations. This accord- 

 ingly was done, by the founders of mechanics, both for the cases of 

 the equilibrium and of motion. Newton's attempt to solve the prob- 

 lem of the figure of the earth, supposing it fluid, is the first example 

 of such an investigation : and this solution rested upon principles 

 which we have already explained, applied with the skill and sagacity 

 which distinguished all that Newton did. 



We have already seen how the generality of the principle, that 

 fluids press equally in all directions, was established. In applying it 

 to calculation, Newton took for his fundamental principle, the equal 



