202 Mr. DE MORGAN, ON DIVERGENT SERIES. 



I now take some results of the two series here discussed which are interesting in the way of 

 verification and extension, though not illustrative of the points on which I am specially writing. If 

 for h and c we write h : a and c : o, we have 



TTCI TTIJ 1 



— — — tto 



?II1} J^ (€*'-€ '" ) COS 



c 



c 



Make a = 0, c = 1 , and we have 



Vsin ■KhI 



1 1 1 



= ... + 7T IT-, + 71 IT-, + r, + 



Vsin Trfi/ 



(6-2)^ (6-1)2 62 (6 + 1)' (6 + 2)' 



1 1 1 l_ 1 



cos x6 = ... + ^^-^, - j^-^, + ^ - ^^^y. + ^j ^ ^y 



(- 1) " d"-^ {I TT \-\_ I _J__ + 1 + __L_ + ^— + 



rra d6"-4Uin,r6J j ■*" (6 - 2)" ■*" (6 - 1 )» 6" (6 + 1)" (6+2)" 



(-l)"d"-|/ ^ X) _! L_ + i__J_ +_JL__... 



ni rf6"-'(Uin^6// "• (6-2)" (6-1)" 6" (6+1)" (6 + 2)" 



■n- 1 1 1 _ 



sinTrft '" 6-2 6-1 6 6+1 6 + 2 



If we had commenced with \{b+pcf - l\~\ and had used the formula j^^t""^"' sin rrfu 



= (1 - »»')"', which Poisson would have admitted as a limiting form of ^^ ^i-k+mV-ii" gjn vdv, we 



should have seen in the final result a right to substitute a\/ - 1 for a in the preceding formulae; 



giving 



. 27ro . irCL 7r6 



sm sin — cos — 



1 IT c c* 2ircc 



*3 



(b + pc)^ - a^ ca Zna 2ir6 (h +pcy - a' ac "wa 27r6 



cos cos cos cos 



c c c c 



Various formulae might be obtained by differentiating these with respect to a, 6, or c ; and 

 various others by integration, one set of which is remarkable. Multiply the two first equations 

 severally by ada, and it will be seen that the second sides become integrable : integrate from a = 0, 

 and make the antilogarithmic change, which gives the following continued products, of which the 

 well-known formulas for the sine and cosine are particular cases. 





^ -?Z2 2ir6 

 ^(e ' + e -■ ) - cos 



27r6 



1 - cos 



c 



x6 Tr6 



C c 



-['■'W^ V^W^- l(e. +e-0 + cos- 1-cos- 



the second of which is readily deducible from the first. It is needless to write down the forms 

 arising from the substitution of a \/- 1 for a. 



