NOTES AND ADDITIONS 



1. GRAPHICAL REPRESENTATION. IDEA OF MATHEMATICAL 



FUNCTION (p. 49). 



The method employed in fig. 16 of representing by a 

 sign the dimensions of the expansion relatively to the amount 

 of the expanding weights, admits of such a variety of appli- 

 cations, and will be used so frequently, that a brief explana- 

 tion of it may not be out of place here. 



When two series of values bear such a relation the one 

 to the other that each value of one series corresponds with a 

 definite value in the other, mathematicians speak of the one 

 value as the function of the other. This relation may 

 always be exhibited in tabular form, as in the following 

 example : 



1234 5 6 7 8 9 10 

 2 4 6 8 10 .12 14 16 18 20 



The relation which prevails in this case is very simple. 

 Each number in the upper series corresponds with a number 

 in the lower, and the latter is always double the value of 

 the former. Representing the numbers in the upper series 

 by x, those in the lower by y, the relation between the two 

 series of numbers may be expressed in the formula : 



y=1x 

 This formula expresses the same and even more than the 



