MK. J. J. WALKER ON THE DIAMETERS OF A PLANK CUBIC. 175 



44. The question of the angle (6) between the polar line of a point (x'yV) on L and 

 the " double " chord of the pencil through that point is of some interest, as a generali- 

 sation of the question of the angle which a Newtonian " diameter " makes with its 

 " ordinates." 



The polar line being 



8u' du' 8u' 



and the double chord (56) 



what may be called the " direction coordinates" of the two lines are (the angles of the 

 triangle of reference being A, B, C) 



and, using A' as in 20 ; A ... H as in 30, 



The angle between two lines whose direction coordinates are X/tj/, X'^V is given, 

 generally, by 



tan 6 = 2 (nv p.'v) sin B sin C/3SXX' sin A cos <4, ... (57) 



and in the particukr case now under consideration 



. . . + 2F^'^+ . . .11 



. . ,)~j I Sx'sin-.-l cos/41 sinC y-, sin^ 5-7) 



' 



(58) 



45. If x"y"z" is the point of contact with its envelope, the poloid *, of the polar 



line of a point x'y'z on L or 



Ix + my + nz = 0, 



from the forms of the equation of that polar line it is evident that 



