from a Vibrating Body to a surrounding Gas. 417 



gral of (15) for integral values of n in the form of a definite 

 integral. 



Let us attend now more particularly to the case of n = 0. The 

 equation (16) is of the form ^ n = Aj\n) -\-Bj\ — n), f(7i) contain- 

 ing r as well as n. Expanding by Maclaurin's theorem, we have 



* =(xV + B)/(0) + (A-B)/'(0)^+(A + B)/"(0) I ^ + . . . 



Writing A for A + B, ji~ ] B for A— B, and then making n vanish, 

 we have 



f =A/(0)+B/'(0), 

 or 



+ =(A + Blogr)(l- 



+ \W l ¥i 2 2 &¥¥ 3 

 where 



S ;i = l- 1 + 2- I + 3- 1 ... +n~\ 



The integral in the form (17) assumes no peculiar shape when n 

 is integral, and we have at once 



{l 2 1 2 3 2 



1 . Smr 1 .2 (8m?-) 2 



■} 



1 2 3 2 5 2 

 ~1.2.3(8i»ir) 8 + 



v ' L l.oimr 1.2(mmry 



1 2 3 2 5 2 



y • • (23) 



1.3.8(8i»ir) 



+ 



•:••}. 



I have explained at length the mode of dealing with such 

 functions, and especially of connecting the arbitrary constants in 

 the ascending and descending series, in two papers published in 

 the Transactions of the Cambridge Philosophical Society*, in 

 the second of which the connexion of the constants is worked 

 out in this very example. To these I will refer, merely obser- 

 ving that while it is perfectly easy to connect A, B with E, F, 

 the connexion of C, D with E, F involves some extremely 

 curious points of analysis. The result of eliminating E, F be- 



* "[On the Numerical Calculation of a Class of Definite Integrals and 

 Infinite Series," vol. ix. p. 166, and " On the Discontinuity of Arbitrary 

 Constants which appear in Divergent Developments," vol. x. p. 105. A 

 supplement to the latter paper has recently been read before the Cambridge 

 Philosophical Society. 



Phil. Mag. S.4. Vol. 36. No. 245. Dec. 1868. 2 E 



