LAPLACE. 515 



9 GO. Laplace extends the method of approximation given in 

 Art. 957 to the case of double integrals. The following is substan- 

 tially his process. Suppose we require \\ydxdx' taken between 



such limits of x and x as make y vanish. Let Y denote the 

 maximum value of y, and suppose that a and a are the correspond- 

 ing values of x and x. Assume , 



y=Ye-''-^\ 



X = a + 6, x = a -\- 6'. 



Y 



Substitute these values of x and x in the function log — and 



expand it in powers of 6 and 6' ; then since Y is by hypothesis the 

 maximum value of y the coefficients of 6 and 6' will vanish in this 

 expansion : hence we may write the result thus 



that is J/ (" (9 + ^ ^'V j^(p-^\ d" =e^ t'\ 



Since we have made only one assumption respecting the inde- 

 pendent variables t and t' w^e are at liberty to make another ; we 

 will assume 



and therefore 6' /(p-^\^t'. 



Now by the ordinary theory for the transformation of double 

 integrals we have 



\ydxdx = jj ^ , 



, T-K , 1 r dt dt' dt dt' 



where x> stands lor -7-, ^7^, — -,^, -77: . 



du do do do 



Thus far the process is exact. For an approximation we may 

 suppose M, N, P to be functions of a and a only ; then we have 



2r da" ' tY dada" 2Y da"' 



O 



3—2 



