SECT. IV.] OTHER FORM OF INTEGRAL. 421 



W = (^) fa fa ^ (*, ft) fa fa COS (px -jpa) COS fe/ - 2/3) 



sin (jp l + g*t) 



If we make t = in u and in IF, the first function becomes 

 equal to $(&,y), and the second nothing; and if we also make 



= in -j-u and in -=- W, the first function becomes nothing, 



and the second becomes equal to ty (x,y) hence v = u + W is the 

 general integral of the proposed equation. 



412. We may give to the value of u a simpler form by effect 

 ing the two integrations with respect to p and q. For this 

 purpose we use the two equations (1) and (2) which we have 

 proved in Art. 407, and we obtain the following integral, 



Denoting by u the first part of the integral, and by W the 

 second, which ought to contain another arbitrary function, we 

 have 



rt 

 TF = 



Jo 



dtu and v = u+ W. 



If we denote by /-t and v two new unknowns, such that we 

 have 



a-x_ * ft-y_ 



;* I7T 



and if we substitute for a, /?, dz, d@ their values 



#4-2^7^, y + 2vji t 2d 

 we have this other form of the integral, 



We could not multiply further these applications of our 

 formulae without diverging from our chief subject. The preceding 

 examples relate to physical phenomena, whose laws were un 

 known and difficult to discover; and we have chosen them because 



