224 PEOCEEDINGS OF THE AMERICAN ACADEMY 



or calling 7"= -^ and X=. x% we have an equation of a linear form, 



T=z m X-\- n. We may then in general write : — 



f=.mf-^n (1). 



Aiid, if one equation can be reduced to this form, we can readily deter- 

 mine whether there are any values of m and n which will satisfy it. 

 Again, if we have : — 



f=rnf^ (2), 



we can reduce to a linear form by using Y= log /and X= log/'. 

 Another common case is : — 



f=mnS' . (3). 



Taking logarithms logy= log m-\- f^ log w, from which log m and 

 log w, and hence m and w, are readily found. The most important 

 application of this formula is when two quantities are so connected 

 that if one varies in arithmetical progression, the other will vary geo- 

 metrically. This is the case for the variations of the barometer for 

 various heights, for the conduction of heat, and the loss of potential of 

 an insulated cable by leakage. In all these cases we have y = m n^, 

 where x varies arithmetically and y geometrically, and from which m 

 and n may be determined as above. 



Probably the .best way of illustrating these principles is by a few 

 examples, in which, however, figures would be required to show the 

 results most clearly. 



1. Torsion Pendulum. Four observations were made on the time 

 of vibration of a torsion pendulum when its length was varied. The 

 results are given in columns 1 and 2 of the adjoining table. The 3d 



TABLE I. 



and 4th columns give their logarithms. On constructing the points 

 with these co-ordinates, they fall very nearly on a straight line, as is 

 shown by column 5, which gives values of log t, computed by the for- 



