136 THEOEY OF HEAT. [CHAP. III. 



so long as the variable y is included between the limits \ TT 

 and + ^ TT. It may be doubted whether such a function exists, 

 but this difficulty will be fully cleared up by the sequel. 



170. Before giving the calculation of the coefficients, we 

 may notice the effect represented by each one of the terms of 

 the series in equation (b). 



Suppose the fixed temperature of the base A^ instead of 

 being equal to unity at every point, to diminish as the point 

 of the line A becomes more remote from the middle point, 

 being proportional to the cosine of that distance ; in this case 

 it will easily be seen what is the nature of the curved surface, 

 whose vertical ordinate expresses the temperature v or fy (x, ?/). 

 If this surface be cut at the origin by a plane perpendicular 

 to the axis of x, the curve which bounds the section will have 

 for its equation v = a cos y ; the values of the coefficients will 

 be the following : 



a = a, Z&amp;gt;=0, c = 0, d= 0, 



and so on, and the equation of the curved surface will be 



v = ae~ x cos y. 



If this surface be cut at right angles to the axis of y, the 

 section will be a logarithmic spiral whose convexity is turned 

 towards the axis; if it be cut at right angles to the axis of x, 

 the section will be a trigonometric curve whose concavity is 

 turned towards the axis. 



It follows from this that the function -7-5- is always positive, 



ctx 



d*v 

 and -^-3 is always negative. Now the quantity of heat which 



a molecule acquires in consequence of its position . between two 



others in the direction of x is proportional to the value of -^ 



ctoc 



(Art. 123) : it follows then that the intermediate molecule receives 

 from that which precedes it, in the direction of x, more heat than 

 it communicates to that which follows it. But, if the same mole 

 cule be considered as situated between two others in the direction 



of y, the function -- a being negative, it appears that the in- 



