254 Dr. 0. J. Lodge on the Variation of the Thermal 



sees that ma, may vary between 4 + 5y ±2 V3(l+2y) without the 

 change from sinh to cosh taking place ; hence the above form is 

 likely to coyer the case of all ordinary metals. 



19. We will denote the constant ^- + J by the letter K, and 



the constant \/R.mct by the letter /x, and will then write the 

 equation (15) in the form 



L , 



-q +J = K cosh ^ + vK 2 + 1 sinh \xx\ . . (17) 



or, what is equivalent, 



^ + J=±^W^ + K)e^-i(VWTl-'K)e-^. (18) 



20. Xow the minimum possible value of L is (this being 



vl + 5y 



its approximate value when w?a=4+5y); and as m is a number 

 likely to be bigger than 0, and as J is always positive and mostly 



greater than 1, it follows that K, or — -{- J, is seldom small; and a 







reasonable value for it is about 5; hence K and Vl+K 2 are ordi- 

 narily not very unequal. For a case when they may be regarded 

 as practically equal the second term of equation (18) vanishes; and 

 it may be written, on this assumption, 



^+J£:E>*, (19) 



u 



or, putting in the value of K, namely - + J, and writing - = A (see 



Li 



equation 16), 



e- ^r (20) 



Amount of Divergence of the corrected Curve from the Loga- 

 rithmic Form. 



21. As the result, then, of the whole investigation, we have 

 the two equivalent equations (17)- and (18), and the approxi- 

 mately equivalent equation (20); which last, however, is only 

 true when K is so large that there is no perceptible difference 

 between K 2 and 1 + K 2 . It remains to see how far these equa- 

 tions agree with the results of the experiments made hitherto, 

 and to show how the conductivity h and the variation-coeffi- 

 cient of conductivity m can be deduced from them. 



Now equation (20) obviously reduces to the ordinary loga- 

 rithmic curve, which was supposed by Biot to represent the 

 curve of temperature along the rod, by making \=0. Equa- 



