412 Mr. Gr. H. Darwin on Variations in the Vertical 



forms, in fact, harmonic mountains and valleys on the infinite 

 plane. We require to find the potential and attraction of such 

 a distribution of matter. 



Now the potential of an infinite straight line, of line-density 

 p, at a point distant d from it, is well known to be —2/xp log d, 

 where //. is the attraction between unit masses at unit distance 

 apart. Hence the potential V of the supposed distribution of 

 matter at the point x, z is given by 



Y = -fywh | + "cos |log */{£ + (Z-zfW 



It is not hard to show that the first term vanishes when taken 

 between the limits. 



XT ± + K~ x ±-i. j. • K • tX Z tX . Z 



Now put t — so that sin T = sin -=- cos T + cos -=- sin 7 > 



1 x b b b b b 



and we have 



V=2pwhb]^ {sin^ cos^ + cos^ sing) j-^p- 



But it is known* that 



C + ™tsmctdt „ C + tD tcosct _ _ 



J.. ■!+?— ^' J_T+?*- a 



Therefore tt o iz _,« 2 



V = 27r/jLwhbe "° cos f • 



If^ be gravity, a earth's radius, and S earth's mean densitv. 



*-* ' j 



"-S--™ i c> 



The deflection of the plumb-line at any point on the surface 

 denoted by x = 0, and z, is clearly dY /gdz. when x=0. There- 

 fore 



1 3grit'A . z 



9 



the deflection = x ■ ' ^ sin ~. . . . (5) 



" lab b v ' 



da. 

 But from (2) the slope (or -=-j when z is zero) is 



•See Todhunter's ' Integ. Calc' ; chapter on •' Definite Integrals." 



