﻿Wire or Tape including the Effect of Stiffness, 109 

 9TTT 



Hence s = c tan ty + w — 2 sin i/r cos 3 i/r, 



which is precisely the value arrived at by Professor Mac- 



laurin, but in deriving this we have neglected altogether the 



complementary function of our differential equation. The 



fallacy here involved is so subtle that the writer wonders 



if a parallel case has occurred in any other physical 



investigation. The only somewhat similar case he has come 



across is in the investigation of the Earth's Precession 



(seeRouth's ' Rigid Dynamics/ vol. ii. p. 325, Ed. 1892), 



but there it is shown that the original neglect of the 



complementary function has not vitiated the result. He 



will think himself fortunate if this paper should induce 



investigators to give more careful attention to the question. 



The effect of the stiffness can be seen more clearly by 



dividing the angle i/r into two parts, one of which <£ is the 



angle due to the ordinary catenary action, and the other 



6 the correction to this due to the stiffness. We have,. 



then, yjr = (p + } 



zcs 

 where <£ = tan _1 ™- =tan -1 fo, 



To 



putting b — w . 



Equation (19) becomes 



^fo~*~ I — — bs cos >Jr + sin ^ = sec (f> sin #, 



T ds 2 



or — ^7-2 — - = & sec 9 sin a ; 



but 



d 2 4> _ W [s_ 



ds 2 ~* (1+W) 1 



and sec<£ = (l + b 2 s 2 )K 



,. g=a'(l+6V)W +(r ^. . . (20) 



This or equation (19) must be solved for the complete 

 general case, but the writer has not been able to find the 

 general solution of either. In the surveyor's tape will 

 always be a very small angle, even with steep slopes and a 

 low tension, so that we may put for sin 6 and solve (20) 

 by approximation. Neglecting b 2 s 2 in comparison with 

 unity we have 



ds" 



