LAPLACE. 511 



The second member may be put in tlie form 







Denote the quantity [ — —Vz^^y by the arbitrary function 

 </) (?/). Thus 



This value of Zx,y will then satisfy the equation in Finite Dif- 

 ferences. 



Each of the n roots q, q^, q^, ... gives rise to a similar ex- 

 pression ; and the sum of the 7i particular values thus obtained for 

 Zx y will furnish the general value, involving n arbitrary functions. 



The student will as before be able to translate this process 

 into the language of the Calculus of Operations. 



Laplace continues thus : Suppose a indefinitely small, and 

 equal to dy. Then 



as we may see by taking logarithms. Thus we shall obtain 



This is the complete integral of the equation 



Laplace next gives some formulae of what we now call the Cal- 

 culus of Operations, in the case of two independent variables ; see 

 his pages 68 — 70. 



954. In his pages 70 — 80 Laplace offers some remarks on the 

 transition from the finite to the indefinitely small ; his object is to 

 shew that the process will furnish rigorous demonstrations. He 

 illustrates by referring to the problem of vibrating strings, and 

 this leads him to notice a famous question, namely that of the ad- 

 missibility of discontinuous functions in the solution of partial dif- 



