CHAP. VII.] THE SOLUTION. 315 



or denoting by 6 1} e 2 , e 3 , &c. the arcs nj., n t l, n 3 l, &c. 



e.y . e 9 y . ey 



sin e, cos -~ sin e 2 cos -~- sin e 3 cos -y- 



1 



+ _ + & c . 



4 2e x + sin e l 2e a + sin e 2 2e 3 + sin e 3 



an equation which holds for all values of y included between 

 and I, and consequently for all those which are included between 

 and I, when x = 0. 



Substituting the known values of a l9 b lt a a , & 2 , a a , b 3 , &c. in 

 the general value of v, we have the following equation, which 

 contains the solution of the proposed problem, 



v _ smnjcosnf fsmnjcoan.y y^~^ , 

 4.4 2 



sin njcosnjs / sin n^cosn.y v^TT^ , &c 

 * in 2?i 2 Z V 2^? + sin 2n^ 



sin w ? cos n.z f sin w.Z cos n. y 



j __ s _ _ _ I _ i ~ g a 



2/i 3 ^ + sin 2n 2 l \ZriJ + sin 2/i^ 

 + &c .................................................... (E). 



The quantities denoted by n lt n^ n B , &c. are infinite in 

 number, and respectively equal to the quantities j , j , , 3 , &c. ; 



the arcs, e 1 , e 2 , e g , &c., are the roots of the definite equation 



hi 

 e tan e = -=- . 



326. The solution expressed by the foregoing equation E is 

 the only solution which belongs to the problem ; it represents the 



general integral of the equation -^ + -^ 2 + y- 2 = 0, in which the 



arbitrary functions have been determined from the given condi 

 tions. It is easy to see that there can be no different solution. 

 In fact, let us denote by -fy (as, y, z] the value of v derived from the 

 equation (E), it is evident that if we gave to the solid initial tem 

 peratures expressed by ty(x, y, z), no change could happen in the 

 system of temperatures, provided that the section at the origin 

 were retained at the constant temperature 1: for the equation 



j-a + -5-5 + ~J~&amp;gt; being satisfied, the instantaneous variation of 

 dx dy dz&quot; 



