506 Geometrical Proof of the principle (/Laplace's Functions. 



angle niP = angle rpY, which ultimately = angle OPD, 

 also 



angle n?r = angle PDO. 



Hence the triangles Pm and DOP are similar to each other; 



DO Yr DP-Dp 



.*. ^ttj = fr = irr-^ ultimately, 



DP P/i mM JJ 



DO.mM_D/;/JI^_ IV 1 1 



•*• ~df dp \j) P dp; - Dp ~ dp ultimatel y- 



Now D0 = c, ?nM = d.Qm = dp, DP 2 =l + c 2 -2^; and by 

 supposing the whole semicircle QP<? divided into equal divisions 

 (each =Pp) at Q' Q". . .p"p'. . . q 1 , and joining DQ'. . . Dp f . . . , 

 adding together all the equations like the above one, and taking 

 the limit, in fact integrating, we have 



C i c^(\-f)dpdf_ r^c 1 (i-c*)d^dp 



= Jo ^"T~L\DQ" DQ7 + \dq'~dcF/ + *" 



+ \D^~D^/ + \D^~DP/ + ■'• " f \Dq 1 ~Wq)j 



= I 2d^ = 47T. 



This simple result (which is remarkable for being independent 

 of c) arises from the second and third, the third and fourth . . . 

 terms destroying each other, leaving only the first and last 

 remaining. 



2. The more general property is now easily proved. At the 

 point Q, where fj! = /n and &)' = &), the value of the function 

 F(///, &)') is ¥{fi, ft>) ; call this F • and suppose F', F", F'". . . F (n) 

 to be the values of F(///, ft)') at the successive points Q', Q ;/ , Q" ; . . . q. 

 Hence, when c=l, 



f )dty dp 



■2cp)'- 



= limit of \ d^~^ ^ m - m )+ F( m , - m ) 



r l C 2 " (1 -c 2 )FQu/, d)dfi' da y' C^C 1 (l-c 2 )F(^',ft)') 

 J-Jo (l+c 2 -2c# " 01 JoJ-i (l+c 2 -2q 



%J0 



+ - +F "' J (^-i)} whcnc = 1 ' 



= limit of 



C 2 * l 

 I df- 



Dq' Dq, 



c 2f y F' — F F"— F' F ( " } 



c + DQ' + J)Q! r ^'" '"TTc 



