20 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



TABLE VI. — Continued. 



(5) The radius of the base of a right cylinder of revolution of height / 

 is a. The centre of the lower base is used as the origin of a system of 

 columnar co-ordinates (r, B, z), the axis of tiie cylinder being the axis of 

 2. A function F, which is continuous everywhere within the cylinder, 

 has the value zero on the curved surface and on the lower base, and the 

 constant value F; on the upper base. The planes 2 = /', z =^ I", divide 

 the cylinder into three portions (1), (2), and (3), in which V is repre- 

 sented analytically by three functions, Vi, V^, V3, respectively. If, when 



z = l', 



.^.. 



and when z = I", ko 



9V. 



h 



9 Vo 

 , -^r—^ , ana wnen c = < , «<, 



' d z a z o' 



hi, k^, ^'3 are given constants, Vi, V.i, V3 are given by the equations 



9V, 



where 



