290 SUPPLEMENT TO CHAP. XTV. [CHAP. m. 



N d of the curve edeik [Fig. 91], we have, as before, y = A N tang. 



d A N = (a z) , or since a = r sin a, z = r sin /3, V = P m . Bm . 

 ki 2 sin a 



from eq. (17), 



_ sin* a sin* ft P 

 2 sin a H r ' 



Inserting the value of H above, we have 



sin" a sin' ft Fl + B K "| B^ 

 2sina Ll-A^jA,' 



(sin* a sin* /3\B t 

 Ssino )A~ ' that is, if yo is the value of y when fc is 



neglected, 



-r=r;' (80) 



which is the value of y given in Art. 159 (2) of the text. In that Art. we 

 have already tabulated the values of A and B, as also of y for various 

 values of a and /3. 



For /3 = a, that is, for the end ordinate, our expression for y reduces to 



. In this case, by differentiating numerator and denominator, we have 



a 3 sin a COS a + 2 (1 + K) a COS 9 a 

 sin a a COS a K (sin a + a COS a)' 



For the semi-circle, a = 90 = , sin a = 1, cos a = 0, and hence 



2 



y = -J TT r = 1.5708 r. Hence, for the semi-circle the intersection curve be- 

 comes a horizontal straight line at 0.5708 r above the crown. In all cases 

 for small central angle a, K may be disregarded. 



The above results are sufficient to enable us to either diagram or calcu- 

 late the strains in every piece for any given position of load. 



