278 THEORY OF HEAT. [CHAP. V. 



Hence the value of each coefficient is expressed thus : 



2 sin n X nX cos n X 



n nX sin nX cos n X 



the order of the coefficient is determined by that of the root n, 

 the equation which gives the values of n being 



nX cos nX ., , v 



: TF = 1 h X. 



sin nX 



We therefore find 



JiX 



a - 



n n X cosec nX cos nX 



It is easy now to form the general value which is given by the 

 equation 



vx e~* Wl2&amp;lt; shifts 



^ 



Denoting by e t , e 2 , e 3 , &c. the roots of the equation 



tan e 



and supposing them arranged in order beginning with the least ; 

 replacing n^X, n 2 X, n Q X } &c. by e^ e 2 , 6 3 , &c., and writing instead 



TT 7 



of k and h their values 7^ and -^ , we have for the expression of 



Ox/ xx 



the variations of temperature during the cooling of a solid sphere, 

 which was once uniformly heated, the equation 



I* C-w xV Ci , 



sm-^F 

 X (. 



X 



K e : x e 1 cosec e x cos e^ 



X 



nn-fe 



+ &c. 



ea) 6 cosec 6 cos e 



Note. The problem of the sphere has been very completely discussed by 

 Biemann, Partielle Differentialglelchungen, 6169. [A. F.] 



