138 THIRD REPORT — 1833. 



all linear partial differential equations of the second order, with 

 constant coefficients, is vahiable as presenting an analogy be- 

 tween arbitrary constants and arbitrary functions. 



But the way in which Lagrange, after estabhshing these two 

 points, proceeds to find the velocity of propagation, does not 

 appear to me equally satisfactory with the rest of his reasoning. 

 His method seems to be a departure from the principle which 

 may be gathered from that of Newton. For, as was mentioned 

 above, the reasoning of the Principia shows that the velocity of 

 propagation is independent of all that is arbitrary. It seems 

 important to the truth of the analytic reasoning, that it should 

 not only obtain a constant velocity of propagation, but arrive at 

 it by a process which is independent of the arbitrary nature of 

 the functions ; whereas the method which the name of La- 

 grange has sanctioned, is essentially dependent on the discon- 

 tinuity of the functions, that is, on their being arbitrary. With 

 a view of calling attention to this difficulty, and as far as possi- 

 ble removing it, the author of this Report read a paper before 

 the Philosophical Society of Cambridge, which is published in 

 Vol. iii. Part I. of their Transactions. I am far from assert- 

 ing that that Essay has been successful; but some service, I 

 think, will be done to science if it should lead mathematicians to 

 a reconsideration of the mode of mathematical reasoning to be 

 employed in regard to the applications of arbitrary functions. 

 If the determination of the velocity of propagation in elastic 

 fluids were the only problem affected by this treatment of arbi- 

 trary functions, it would not be worth while to raise a question 

 respecting the principle of the received method, as no doubt 

 attaches to the result obtained by it ; but there are other pro- 

 blems, (one we shall have to consider,) the correct solutions of 

 which mainly depend on the construction to be put upon these 

 functions. The difficulty I am speaking of, which is one of a 

 delicate and abstract nature, will perhaps be best understood 

 by the following queries, which seem calculated to bring the 

 point to an issue : — Can the arbitrary functions be immediately 

 applied to any but the parts of the fluid immediately acted upon 

 by the ai'bitrary disturbance, and to parts indefinitely near to 

 these ? To apply them to parts more remote, is it not necessary 

 first to obtain the law of propagation ? And do not the arbi- 

 trary functions themselves, by the quantities they involve, fur- 

 nish us with means of ascertaining the law of propagation, 

 independently of any consideration of discontinuity? 



Euler and Lagrange determined the velocity of propagation 

 in having regard to the three dimensions of the fluid, on the li- 

 mited supposition that the initial disturbance is the same as to 



