134 PROCEEDINGS OP THE AMERICAN ACADEMY 



The results of our work on " longitudinal curves " are given below. 

 Evidently, iu crossing the plate, we shall find, in general, four points 

 having the same temperature, two about either pole. The isothermal 

 .points lying about one pole iu each of two experiments, obtained by 

 laveraging several readings of the curves mapped out, are as fojlows: — 



The isothermal curves corresponding to these pairs of points are in 

 general form like the family to which tlie common lemniscate belongs. 

 Several other curves have much the same form, and therefore it be- 

 comes necessary to show that our experimental curve is none of these. 

 The foUowing conditions must be satisfied : — 



(1) For a given positive value of r^ tliere must be only one real 

 and positive value of r^, r^ and r^ being radii from the fixed points. 



(2) The function must be homogeneous ; else, for the same physi- 

 cal problem we shall get as many diflfereut solutions as we employ 

 units of measure. 



Hence we must' exclude trigonometric functions, by reason of their 

 periodicity, and, since exponential and logarithmic forms may be devel- 

 oped in algebraic series, need consider only these last. 



In this system of coordinates, negative, as well as truly imaginary 

 values of r^ and r^, are to be regarded as imaginary. (1) requires us to 



