108 APPLICATION OF THE PRELIMINARY RESULTS 



In precisely the same way the value of ty is found to be 







= - 



It now only remains to find the values of X and Y in functions 

 of x. By supposing p placed on the axis of the wire, the equa- 

 tion (c) becomes 



F= ~JMWW ; 



the integral being extended over the whole surface of the wire : 

 Y' belonging to the point p\ whose co-ordinates will be marked 

 with an accent. 



The part of Y' due to the circular plane at the end of the 

 cylinder, where x = X, is 



dX" [ a 27rrdr dX" 



71 /7V"" 



since here da- = %7rrdr and ~ = -^ , by neglecting quantities 



of the order a 2 on account of their smallness ; X" representing 

 the value of X when x = X. 



At the other end where x + X we have da- = 2irrdr 9 



d$ = dX'" 

 dw dx J 



and consequently the part due to it is 



dX' 



X'" designating the value of X when x = + X. 

 At the curve surface of the cylinder 



, .3$ d$ l d*X 

 da = %7radx and -f = -f- = \a j-s- , 

 dw dr dx 2 



provided we omit quantities of the order a 2 compared with those 

 retained. Hence the remaining part due to this surface is 



, [dx d*X 

 Tra I -r^: j-s- ; 

 J (r) dx* ' 



