REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 137 



conduct to the same velocity of propagation. This, which was 

 thought to be an objection to the reasoning, is an evidence of 

 its correctness : for the plain consequence is, that the velocity 

 of propagation is independent of the kind of vibration which 

 we may arbitrarily impress on the fluid ; — and so experience 

 finds it to be. 



When the partial differential equation, which applies equally 

 to the vibrations of the air and those of an elastic chord, had 

 been formed and integrated, a celebrated discussion arose 

 between Euler and D'Alembert as to the extent to which the 

 integral could be applied ; whether only to cases in which the 

 motion was defined by a continuous curve, or also to motion 

 defined by a broken and discontinuous line. It is well known 

 that the question was set at rest by Lagrange, in two Disser- 

 tations published in vol. i. and vol, ii. of the Miscellanea Tau- 

 rinensia. The difficulty that arrested the attention of these 

 eminent mathematicians was one of a novel kind, and peculiar 

 to physical questions that require for their solution the integrals 

 of partial differential equations. The difficulty of integration, 

 which is the obstacle in most instances, had been overcome by 

 D'Alembert. It remained to draw inferences from the inte- 

 gral, — to interpret the language of analysis. When an aggre- 

 gate of points, as a mass of fluid or an elastic chord, receives 

 an arbitrary and irregular impulse, any point not immediately 

 acted upon may have a correspondent irregular movement after 

 the initial disturbance has ceased. This is a matter of experi- 

 ence. Was it possible, then, that these irregular impulses, and 

 the consequent motions, were embraced by the analytical calcu- 

 lation ? From Lagrange's researches it follows that the func- 

 tions introduced by integration are arbitrary to the same degree 

 that the motion is so practically, and that they will therefore 

 apply to discontinuous motions. (Of course we must except 

 the practical disturbances which the limitations of the calcula- 

 tion exclude, — those which are very abrupt, or very large.) 

 This has been a great advance made in the application of ana- 

 lysis to physical questions. Had a different conclusion been 

 arrived at, many facts of nature could never have come under 

 the power of calculation. The Researches of Lagrange, which 

 will ever form an epoch in the science of applied mathematics, 

 establish two points principally : First, That the arbitrary func- 

 tions, as we have been just saying, are not necessarily conti- 

 nuous : Secondly, That (in the instance he considered) they are 

 equivalent to an infinite series of terms having arbitrary con- 

 stants for coefficients, and proceeding according to the sines of 

 multiple arcs. This latter result, which appears to be true for 



