312 THEORY OF HEAT. [CHAP. VII. 



tity y. Now, it is easy to see that there are an infinite number 

 of arcs which, multiplied respectively by their tangents, give the 

 same definite product -j-, whence it follows that we can find 



K 



for n or p an infinite number of different values. 



322. If we denote by e lt e 2 , e a , &c. the infinite number of 

 arcs which satisfy the definite equation 6 tan e = ^- , we can take 



for n any one of these arcs divided by I. The same would be the 

 case with the quantity p ; we must then take w 2 = n 2 + p 2 . If we 

 gave to n and p other values, we could satisfy the differential 

 equation, but not the condition relative to the surface. We can 

 then find in this manner an infinite number of particular values 

 of v, and as the sum of any collection of these values still satisfies 

 the equation, we can form a more general value of v. 



Take successively for n and p all the possible values, namely, 

 ^, -j, ^ 3 , &c. Denoting by a lf a 2 , a 3 , &c., 7&amp;gt; 1? 6 2 , 6 8 , &c., con 



stant coefficients, the value of v may be expressed by the following 

 equation : 



v = (a l e~ x % 2 +% 2 cos njj -f a a e&quot; a ?+^ cos njj + &c.) \ cos n^z 

 4- (a^e~ x ^ + n ** cos n$ -f a -* ****+&quot;* cos njj + &c.) 5 2 cos n^z 

 + (a^-* V ^ 2+W 3 2 cos n 4- af-****+* cos n z y -f &c.) b a cos n 3 z 

 + &c. 



323. If we now suppose the distance x nothing, every point of 

 the section A must preserve a constant temperature. It is there 

 fore necessary that, on making x 0, the value of v should be 

 always the same, whatever value we may give to y or to z ; pro 

 vided these values are included between and I. Now, on making 

 x 0, we find 



v = (a t cos n^y + a 2 cos n z y + a 3 cos n 3 y + &c.) 

 x (^ cos n^z 4- &amp;gt; 2 cos n z y -f & 3 cos n z y + &c.). 



