63 



" On the Equations and on some Properties of Projected 

 Solids," by James Bottomley, D.Sc, B.A., F.C.S. (abstract). 



On a former occasion I brought before the Physical and 

 Mathematical Section of this Society a proposition in pro- 

 jection, in which it was shown how by the composition of 

 two projections, namely, of that of a line on a line, and of 

 that of a plane area on a plane area perpendicular to the 

 aforesaid line, we could derive from a solid three solids with 

 axes perpendicular to three planes and of variable volume ; 

 the variation being subject to the condition that the sum 

 of the three volumes is constant and equal to that of the 

 primitive soKd. I now propose to solve the following 

 problem, given the equation to the primitive solid to deduce 

 that of a derived solid. 



Let tlie equation to the primitive solid refeiTed to three 



rectangular areas be 



F(x, y, z) = 0. 



Let ABC be the pri- 

 mitive plane which is 

 fixed in the solid, and DE 

 m axis perpendicular to 

 bhis plane, and which may 

 be called the primitive 

 ixis. Let P be a point 

 situated on the intersec- 

 tions of the solid by a 

 plane parallel to the 

 primitive plane. Draw 



PG perpendicular to the 

 plane x, y; on PG take a length LG so that 



LG - PFcosy 

 7 being the inclination of the primitive axis to the axis of z. 

 Then L will be a point on the derived solid. Also we have 



PF = PDcosDPF 

 DPF is the angle between PF and PD, PF is parallel to the 

 primitive axis, and its direction cosines will therefore be 



