602 Dr. C. Chree : Applications of 



be respectively replaced by o— Vft, the positive root of 

 £ 3_ 9 .2 + 3 ^ + i = o, and VI/3. 



§ 25. Every element o£ the upper surface is stretched in 

 Case (i.) . Thus it is impossible to keep the length of any 

 finite portion of the upper surface unaffected unless a/l exceed 

 0"5. By altering a/l we can secure that any symmetrical 

 portion we choose of the upper surface shall have its length 

 unaffected. We have only to make the slope nil at the end& 

 of the portion considered. 



Denoting the symmetrical length to be kept unaffected by 

 2b, we have the position of the supports given by one or other 

 of two equations, viz. : — 



«/'=HKW. (80) 



when b/l lies between and '5505 ; 



"/KHi-a-w}]*, .... (si) 



when b/l exceeds '5505. 



Any given arrangement of the supports secures the un- 

 changeability of only one definite symmetrical portion of the 

 length. We may, however, have unchangeability of length 

 in an asymmetrical portion. Supposing the unchanged 

 length wholly included between the centre and a support, its 

 terminal abscissae being f x and f 2 , we find by equating the 

 terminal values of dy/dx, 



£i 2 + fif 2 + ? 2 2 = 3(2«Z-P) (82) 



— 3#i 2 , or x 2 2 by (76) and (74). 



Regarding f 2 and f 2 , and so x^ as known, we determine 

 the position of the supports from the equation 



«/^=i+i(fi 2 +fi&+& 2 )/P = i{l + (W0 2 }-- • (83) 



This applies only when a/l exceeds 05. 



Answering to any given value of a/l exceeding 0'5, w r e 

 have an infinite number of pairs of values f 1? £ 2 . The ends 

 of the asymmetrical portion necessarily lie on opposite sides 

 of the point x = x 1 where the slope is a maximum. 



In cases (iii.), (iv.), and (v.) we can have unchanged length 

 in an asymmetrical portion whose ends lie on opposite sides 

 of a support, though still on the same side of the centre. 

 The abscissae of the extremities of this portion being f t and 

 £ 2 , the corresponding position of the support is given byj |j 



<*=Zi+{(i-&r-(i-mHzo-*, ■ ■ (84> 



or h= Vl -( Vl S- v W(3l)~i, (84'> 



where b=l-a, W =I-fi, % = *-&. 



