﻿1(3 Prof. L. N. G. Filon : Investigation of Stresses 



Consider separately the cases where q > 0, q < respectively. 

 It will be convenient to discriminate briefly between the state 

 of things in a sector in which every radius vector meets the 

 curve in real points, and one in which a smaller sector is 

 contained, in which there are no points of the curve. We 

 shall speak of the first as a loop, of the second as a fork. 

 These expressions are suggested by the forms shown in figs 5 

 aud 6, but they are not to be regarded otherwise than as a 

 convenient notation. 



Case /., q>0. For real values ^?/ 2 sin 2i >u. Now in the 

 sector EOF (fig. 3) u is zero at the boundaries of the sector 

 and negative throughout. Thus real values exist and we 

 have a loop. The reasoning on p. 14 then shows that every 

 radius vector meets the curve in one real point in the field of 

 view (see figs. 5, 6, and 7). 



In the sector DCE, ?t = at the boundaries and is positive 



throughout, having a maximum of 4sin 3 ( — - — ). Hence, 



'. • 3 7T-\-2i \ o -J 



according as qb' 2 sin 2i^4 sin 3 — ~^ we have a loop of the 



o 



type shown in fig. 6 or a fork of the type shown in fig. 5. 



We have, however, to show that qb' 2 is actually capable of 

 taking up values that will ensure the occurrence of the two 

 cases. 



"We have 



q ~\ b +4WW/V h<i + 4W6V 

 _ r4(H -H,) 



{ «itik) +s ,}7, 



For positive values of q (q is clearly capable of taking all 

 possible values, by varying M ) this has a minimum value 

 zero when q = 4 : (R 1 — R )/b 2 . For values of q smaller than 

 this, b'<0 and we have then seen that no part of the curve 

 lies in the field of view. 



\\ e note also for future reference that for negative values 

 of q there is an algebraic maximum when 



7 =-4(H 1 -H )/6«, 



giving 



?6'«=16(H -.H 1 )=-2-91. . . . (16) 



Leaving such negative values of q out of consideration for 

 the present, it appears that qb' 2 sin 2i can be made to range 

 from to +oo. It can therefore be made either greater 



