of the Sums of Neutral Series. 14-1 



we likewise deduce the fact that 



— -f cos + cos 2 + &c. in inf. 

 2 



1 fcos(rc + 1)0 — coswH 



>n = oo. 



2sin 2 — 



L 2(1 -cos 9) 



Prof. Young indeed " considers it an axiom, that what holds 

 for all but the extreme case will hold for that too," but I 

 must beg to submit that this is a matter of fact and not of opi- 

 nion; and the fallacy of the principle in the present case I 

 have sufficiently shown in the former part of this paper. With 

 all due deference therefore to Prof. Young, I shall reassert, 

 that Mr. De Morgan is in error in affirming (1.) to be the 

 limit of the proposed series "when A = 1." Omit the words 

 "when A = 1," and I admit the proposition. Insert those 

 words, and the fact expressed is untrue, if it be not wholly 

 unmeaning. 



Prof. Young says again, " It is easily proved that 



z* 00 a Z* 50 . 1 



/ e~ ax o.o%xdx = 5, / e- ax smxdx=.— i — 5, (a.) 



Jo l+a 2 »/o l-fa 2V/ 



from which it certainly follows, though the inference is denied 

 by Mr. Moon, that in the limit, when a = 0, the true values 

 of these integrals are and 1." I do not deny the inference, 

 that the limits of the integrals when a is diminished indefi- 

 nitely, so long as it continues an actual magnitude, are and 

 1 ; but I do deny that the limits of the integrals are to be 

 found by putting a = 1 in the left-hand members of the two 

 equations («.), that is, I deny that 



/ co&xdx and / sin#d.r 



<J *J 



are the limits of 



/ e-™ cosxdx, I e-^svbxdx, 



which is all I care to establish. 

 December 10, 1845. 



Postscript 2. 



I have read Prof. Young's third paper. Had my reply to 

 his first paper been inserted in proper course, it is probable 

 that he would have saved himself the trouble of writing, and 

 the public of reading his last two papers. The staple of what 

 I have to advance in opposition to this last is contained in my 

 two previous notices. A few words, however, are still called 

 for. 



