Mass of Air confined by Walls at Constant Temperature. 323 



the summation extending to all the values of m in (42). 

 Since (for each term) the mean value of s is zero, the right- 

 hand member of (51) represents also #/©, where 6 is the 

 mean value of 0. 



If in (51) we suppose /3=0, we fall back upon a known 

 Fourier solution, relative to the mean temperature of a 

 spherical solid which having been initially at uniform tempe- 

 rature ® throughout is afterwards maintained at zero all over 

 the surface. From (42) we see that in this case sin m is small 

 and of order ft. Approximately 



sin m=3aft/m ; 



and (51) reduces to 



of which by a known formula the right-hand member iden- 

 tifies itself with unity when £ = 0. By (9) with restoration 

 of a, 



/ i= (l 2 , 3 2 , 5 2 , ....>7r 2 /a 2 (53) 



In the general case we may obtain from (42) an approxi- 

 mate value applicable when m is moderately large. The 

 first approximation is m = iir, i denoting an integer. Suc- 

 cessive operations give 



. t 3a ft 18* 2 /3 2 + 9* 3 /3 3 ,-.. 



™=^+ 1 - ^ • • (54) 



In like manner we find approximately in (51) 



sin 2 m (l+aft)/*ft _ 6(l + *g) f , _ 15*/3 + 9* 2 / 3 2 V 



i 2 7r 2 y 



.... (55) 



showing that the coefficients of the terms of high order in (51) 

 differ from the corresponding terms in (52) only by the factor 

 (1 + aft) or 7. 



In the numerical computation we take 7 = 1'41 , a/3 = '41. 

 The series (54) suffices for finding m when i is greater than 2. 

 The first two terms are found by trial and error with trigo- 

 nometrical tables from (42) . In like manner the approximate 

 value of the left-hand member of (51) therein given suffices 

 when i is greater than 3. The results as far as / = 12 are 

 recorded in the annexed table. 



. , 'daftf- sin 2mA 



