LAPLACE. 559 



Let X = — ^ , r = nr, z^^^ = Z7; then lie saj^s that neglecting 

 terms of the order —^ the equation becomes 



It is difficult to see how Laplace establishes this ; for if we adopt 

 his expressions for 2;j.^i,r, ^x-i,/? and z^^^^^-^, the equation becomes 



dr \ n) \ n) dp 



/, 11^ 4 4\ d^U 



and thus the error seems to be of the order -, or even laro^er, since 



n ° 



p^ may be as great as n. 



1000. Laplace proceeds to integrate his approximate equation 

 by the aid of definite integrals. He is thus led to investigate some 

 auxiliary theorems in definite integrals, and then he passes on to 

 other theorems which bear an analogy to those which occur in 

 connexion with what are called Laplace s Functions, We will give 

 two of the auxiliary theorems, demonstrating them in a way which 

 is perhaps simpler than Laplace's. 



To shew that, if i is a positive integer, 



/» 00 /• 00 



/ e-''-'-'{s-{-fiJ~^)'dsdtJL = 0. 



*' -00 "^ -00 



Transform the double integral by putting 



s = r cos 6, fju = r^m.B\ 

 we thus obtain 



[ * ['" e-"' (cos ^ 6> + ^/^ sin i 6) r'^' dr dO. 



It is obvious that the positive and negative elements in this 

 integral balance each other, so that the result is zero. 



Again to shew that, if t and q are positive integers and q less 

 than ij 



