854 Mr. A. Cayley on the Theory of Logarithms, 



equation, afterwards the temperature Tg from the third, and 

 lastly, we employ all these to determine the work from the first 

 equation. 



In doing so, however, we encounter a peculiar difficulty. In 

 order to calculate T, and Tg from the two last equations, they 

 ought in reality to be solved according to these temperatures. 

 But they contain these temperatures not only explicitly, hut 

 implicitly, p and g being functions of the same. If, in order to 

 eliminate these magnitudes, we were to replace p by one of the 

 ordinary empirical formulae which express the pressure of a vapour 

 as a function of its temperature, and g by the differential coeffi- 

 cient of the same, the equations would become too complicated 

 for further treatment. We might, it is true, like Pambour did, 

 help ourselves by instituting new empirical formulae more con- 

 venient for our purpose, which, if not true for all temperatures, 

 would be correct enough between certain limits. Instead of here 

 entering into such experiments, however, I will draw attention 

 to another method, by which, although the calculation is some- 

 what tedious, the several parts of the same are capable of easy 

 execution. 



[To be continued.] 



XLIII. Second Note on the Theory of Logarithms. 

 By A, Cayley, Esq,^ 



THE theory of logarithms, as developed in my first note (Phil. 

 Mag. April 1856), may be exhibited in a clearer light by 

 considering, instead of log a + log 3 — log ai = E7n, the equation 



log— =log6 — log« + E7ri, a form which more readily enables 



the accounting a priori for the discontinuity in the value of E. 

 Writing then for b, a the complex values oJ + y*, x + yi, we have 



log-^^=log (^4-/i)-log (ar-hy«) +E7ri; 

 x-ryi 



where, according to the assumed definition of a logarithm, 

 log {pB + yi) = log \^x^ + y^ + i ( tan" ^ - + evr J, 



in whicn Idg V'a?*-!-^* is the real logarithm of \/a?^-f-y*, and 



tan~* — is an arc between the limits — tt* + tt^ The coefficient 

 X 3 2 



e is equal to zero when x is positive ; but when x is negative, 



then esE-hl or —1, according as y is positive or negative, 



^0«Si'Vii^>ii.|;iyiliOommunicated by the Author. 



