236 Wisconsin Academy of Sciences, Arts, and Letters. 



be placed upon it and propelled perpendicularly against the ver- 

 trical plane by a blow, which takes effect above its center of grav- 

 ity. Such a blow will impart to the ball a rotary motion, together 

 with an onward motion or translation. When the ball reaches 

 the vertical plane its rebounding force, due to translation, will tend 

 to make it retrace its path, while the force due to its rotation will 

 tend to make it climb the vertical plane. It is actuated by the 

 resultant of these two forces, and rebounds through the air, in 

 the plane of those forces following the diagonal of the rectangle 

 of forces. 



The following diagram* may serve to make the explanation more 

 apparent: Let A, B, C, D, be the vertical plane ; C, D, E, F, the 

 horizontal plane ; 

 Let a be the point 

 from which the 

 ball d is propelled 

 on a-h ; the ball 

 having a forward 

 rotary motion; h-d 

 the distance the 

 ball would re- 

 bound by virtue -^ 

 of its rectilinear motion ; h-c the distance it would climb by vir- 

 tue ot its angular motion. Then will it be found somewhere on 

 the Ime h-e. Being a rectangle of forces, the resultant may be 

 expressed by the formula h-e = V{b-cy + (b-dy. 



If the ball is propelled from a point to the right of its center 

 of gravity, and constrained to keep the same perpendicular course, 

 It will have a negative or left-hand rotation ; when it strikes the 

 vertical plane it will not return in the same path, but will be re- 

 flected to the right, so that the angle of reflection is not equal to 

 the angle of incidence. But just as before, the path of the re- 

 turning ball is the resultant of two forces acting at right angles 

 to each other. If the angular velocity is very great, compared 



*No cuts having been furnished by the author, the printer has been obliged to construct the 

 accompanying figares, which are necessarily very imperfect. 



