and its Application to the Molecular Motions of Gases. 43 



§ 8. In order now to form those differential coefficients of k, h 9 

 and A— w with which we have to do in the theorem of the mean 

 ergal (of which we select h for consideration first), we must in- 

 quire in what relation the two partial differential coefficients ob- 

 tained when h is represented as a function u and c stand to those 

 which are obtained when h is exhibited as a function of iu and c. 

 I wish, however, to premise a few remarks on the notation. 



In cases like the present, where a quantity is represented as 

 a function of two variables, while, from the nature of the subject, 

 we do not always employ the same two, but sometimes change 

 the variables, and where, consequently, partial differential coeffi- 

 cients occur which only differ from one another by the fact that 

 the quantity which in the differentiation was presupposed con- 

 stant is different in one of them from what it is in another, it is 

 convenient to indicate this in the formula, in order that it may 

 not be necessary always to state it verbally. I have therefore, 

 in a previous memoir*, used a notation which is also adopted by 

 various other authors in treating the same subject; that is, I 

 have added as an index the quantity which in the differentiation 

 is regarded as constant. The external form, however, in which 

 this was done can be simplified if, instead of putting the index 

 with the entire differential coefficient (in which case the latter 

 must be enclosed in brackets, and also confusion may arise with 

 other indices possibly occurring in this place, unless the index 

 be furnished with a distinguishing mark), we put it with the d 

 in its numerator ; and this is the form in which we will here 

 employ that notation. 



If, for example, the quantity h is regarded, first, as a function 

 of u and c and so differentiated according to c (when u is con- 

 sidered a constant), and, second, as a function of w and c and 

 so differentiated according to c (when w is considered a con- 

 stant), we write the two differential coefficients thus, 



dji oy _ -, d w h 



-=— and — r— • 

 dc dc 



This way of writing them agrees with that which I employed, 

 in the memoir " On a new Mechanical Theorem relative to Sta- 

 tionary Motions "-f, for the specializing of variations, as I put 

 the measuring quantity (which is regarded in the variation as 

 constant) as an index to the 8 J. 



* Pogg. Ann. vol. exxv. p. 368; Abhandhmgensammlung, vol. ii. 

 p. 14. 



t Sitzungsb.d. Niederrhein. Ges. fur Natur- und Heilkunde, 1873; Phil. 

 Mag. S. 4. vol. xlvi. pp. 236, 266. 



+ Just as my memoir was ready for the press, I received the just pub- 

 lished new Part of the Fortschritte der Physik, and observed in it that 

 Boltzmann also has simplified the form of the index-furnished differential 



