310 MR. L. N. U. FILON ON THE RESISTANCE TO 



throughout the section ; and 



dw/dn + (w^ 5 ty} T = 



along the boundary, where dn = an element of the outwards normal to the boundary, 

 r is the angle of torsion per unit length, and I, m are the direction-cosines of dn. 



Now in the present case 



dn = d X (c v/J) 

 where 



at the boundary where f = const., and 



dn = dr) X (c v/J) 



tit the boundary where TJ = const., the sign being so determined that dn is positive. 

 By adding suitable terms to w, we can reduce one or other of the boundary 



conditions to the form 



dwifdn = 0, 



where 



iv w>, + suitable terms. 

 Suppose we make 



Expanding now Wi in the form of a series, 



"=" . . , f2+ 1 , 

 y, = 2 A, smh < - - TT (77 + K ) \ sn 



n=0 I * 



u>, = i A, smh x ir(t) + K)\ sin 



n=0 I ^ J - 



the differential equation and the first boundary condition are identically satisfied. 

 When this value is substituted in the second boundary condition, we get an 



equation expressing a given function of f in a series of sines of odd multiples of --, 



between the limits + a and a. 



But such an expression can be definitely obtained by a method analogous to that 

 for FOURIER'S series. Comparing coefficients, we obtain relations which determine 

 completely all the constants in the expression of w v 



w is then known. The shears and torsion moment are then deduced by differen- 

 tiation and a double integration. 



2. Summary of the Results. 



The cross-sections which are dealt with in the present paper are of very great 

 generality, and they include as special cases many of the cross-sections which SAINT- 

 VENANT has worked out, for instance the rectangle and the sector of a circle. 



The first section of which I treat is that bounded by an ellipse and two confocal 



