22 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Ai s' = A^s' + B^ c, Jc^ Ai c' — kz {A^ c' + B^ s'), 



A^ s" + B., c" = As s" + Bs c", yC-2 A^c" + k^B^ s" ^ k^ A^ c" + t, B^ s", 



AsS + BsC = il . 



The determinant of the coefficients of the s's and c's may be reduced 

 to the form 



c's'iki-k^) kic'^-k^s'^ 



c s" s c" — c s" c c" 



k^ c c" ks {s s" — c c") kz c s" 



(30) 



and 



A, = (31) 



— ko Ag c n 



c's'{ICi—k2) \ k-2CS"{sc"—CS")+k^Cc"(cc"S!i") j ■j-(kiC"^k<^'^) { Cs"/g(.ss"-Cf ")+^2Cc"(s"c-Ac") I 



If in the special case where ^i and k^ are equal, we write ki = /x k^ = k.^ , 

 we get 



A, = (32) 



-Mfl 



c's'(fi. — l)\s"(sc" — cs") + fxc" {cc" — ss")\ + {fjLc''^ — s"^) \fj.s"(ss"—cc")+c"{s"c — sc")\ 



with corresponding values for the other coefficients. 



We shall need at the outset only two or three applications of the fore- 

 going theory. We may ask first what must be the relative dimensions of a 

 homogeneous regular right prism, one end of which is kept at the uniform 

 temperature ^o ^I'J the other end at the uniform temperature 6„ while 

 its other faces are kept at some uniform temperature 6, between 6q and 61, 

 in order that the tem{)eratures on the axis of the prism in the final state 

 shall be sensibly the same whatever value has. If, for instance, the 

 difference between 60 and 61 is 100° C, what must be the ratio of the 

 radius a of the circumference inscribed in a right section of the prism to 

 the height I of the prism, in order that the temperature of every point on 

 the axis may be the same within less than 0°.01 C, whether 6 is ecjual 

 to 60 or to 61 ? Since we need merely to find a lower limit for a -^ /, 

 we shall do well to substitute for the prism the inscribed right cylinder 

 of revolution, and then apply the solution of Problem 4 given above. 

 We are to find a function of r and z, harmonic for values of r between 

 and a and values of z between and I, which (1) has the uniform 

 value F (Oo) when z — 0, whatever ?• is ; (2) has the uniform value F (0,) 



