SECT. I.] STEADY MOVEMENT IN A RING. 89 



107. Suppose a portion of the circumference of the ring, 

 situated between two successive sources of heat, to be divided 

 into equal parts, and denote by v lt V 2 , V 3 , v 4 , &c., the temperatures 

 at the points of division whose distances from the origin are 

 x v x v x v #4&amp;gt; & c -j the relation between v and x will be given by 

 the preceding equation, after that the two constants have been 

 determined by means of the two values of v corresponding to 



Ju 

 the sources of heat. Denoting by a the quantity e KS , and 



by X the distance x 2 x^ of two consecutive points of division, 

 we shall have the equations : 



whence we derive the following relation - * = a x + a~ A . 



We should find a similar result for the three points whose 

 temperatures are v 2 , v s , v 4 , and in general for any three consecutive 

 points. It follows from this that if we observed the temperatures 

 v \&amp;gt; v v v s&amp;gt; v v V 5 & c - f several successive points, all situated between 

 the same two sources m and n and separated by a constant 

 interval X, we should perceive that any three consecutive tempe 

 ratures are always such that the sum of the two extremes divided 

 by the mean gives a constant quotient a x + a~ A . 



108. If, in the space included between the next two sources of 

 lieat n and p, the temperatures of other different points separated 

 by the same interval X were observed, it would still be found that 

 for any three consecutive points, the sum of the two extreme 

 temperatures, divided by the mean, gives the same quotient 

 k*. 4. a -\ The value of this quotient depends neither on the 

 position nor on the intensity of the sources of heat. 



109. Let q be this constant value, we have the equation 



V s $.-; 



we see by this that when the circumference is divided into equal 

 parts, the temperatures at the points of division, included between 



