188 On Taylor's Theorem. 



termined*, the constant quantity m must be such as to rentier 



— the same for the same intensity of radiation. 



m " 



In order to determine the value of m for any instrument, 

 suppose, while the intensity of the sun's rays continues un- 

 changed, v to be the velocity of heating at the temperature t, 

 and v 1 at some higher temperature t\ there is then the equa- 

 te t/ 



tiona'H = a v + — , by means of which is found 



m m J 



v—x! 



m = — ; ;. 



IV 



When m has been determined for any scale, if m degrees 

 of that scale be adopted as the unit of a new scale, it follows 



that V degrees of this latter scale will be equivalent to — de- 

 grees of the other, and equation (2.) will, for all instruments, 

 be a 1 + V = a x . . . . (4.). 



If t and x are degrees of the centigrade scale, the value of 

 a, according to Dulong and Petit, is 1*0077. But as -0077 is 

 a small fraction, and t, either when positive or negative, sel- 

 dom exceeds 44, it follows that a* = (1 + *0077') may be ex- 

 pressed with sufficient accuracy by the first three terms of its 

 development, and these being substituted in equation (4.), it 

 becomes 



1 + -00767 xt+ -000038 xt*+ V = (1-0077)*. 



But if the temperature be expressed in degrees of Fahren- 

 heit's scale, then a is equal to 1*00425, and the equation be- 

 comes 



1 + -00424 x T + -0000092 x T 2 + V = ( 1 -00425)*. 



As the temperatures to which these formulae relate are 

 comparatively small, it is of no importance whether they be 

 measured by the air-thermometer or by one of another kind. 

 Edinburgh, Feb. 20, 1835. 



XXII. On Taylor's Theorem. By A Correspondent. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



T^HAT every proof which can be given of this theorem 

 -■• will meet with objections may be suspected from the cir- 

 cumstance of every one which has hitherto been given having 



• See Professor Powell's M Report on the present state of our know- 

 ledge of the science of radiant heat," (in the Second Report of the British 

 Association,) under the head " Measures of radiation." 



