On the Inversion of the Independent Variable. 



535 



of weight of liquid water) by dividing by Joule^s equivalent of 

 the specific heat of liquid water^ which has the following values, 

 according to the units of temperature and length employed. 



Logarithms. 

 Centigrade scale, and metres . . 423-54 2-6268969 

 Centigrade scale, and feet . . 1389-6 3-1428898 



Fahrenheit's scale, and feet . . 772-0 2*8876173 



Constants in the Formula for Pressures in millimetres of Mercury, 

 and Temperatures in Centigrade degrees. 



To adapt the formulae to other scales of pressure, add the fol- 

 lowing logarithms to the constants A :- — 



For inches of mercury 2*59517 



For kilogrammes on the square metre . . 1*13341 

 For pounds avoirdupois on the square foot . 0*44477 



To adapt the formulae to the scale of Fahrenheit's thermo- 

 meter, multiply B by 1*8, and C by (1*8)^=3*24; that is to aay, 



Add to log B . . . 0*2552725 

 Add to log C . . . 0*5105450. 



LXIV. Note on Burman's Law for the Inversion of the Indepen^ 

 dent Variable. By J. J. Sylvester, F.R.S.^ 



THIS Note refers to the development of the nth dififerential 

 coefficient of u in respect to x in terms of the nth and 

 lower differential coefficients of £c in respect to u. 



The late Mr. Gregory, in his very valuable book of examples 

 on the Calculus, in alluding to this development, speaks of it as 

 "extremely complicated, and involving so much preliminary 

 matter for its demonstration/^ that he contents himself "with re- 

 ferring to a memoir by Mr. Murphy on the subject in the Phi- 

 losophical Transactions, 1837, p. 210.'^ The development there 

 given is of course essentially no other than that included in Bur- 



* Communicated by the Author. 



