1878.] Mr. W. H. L. Russell on certain Definite integrals . 129 


February 28, 1878. 

Sir JOSEPH HOOKER, K.C.S.L, President, in the Chair. 

The Right Hon. Sir William Henry Gregory was admitted into the 
Society. 

The Presents received were laid on the table, and thanks ordered 
for them. 

The following Papers were read:—• 


I. u On certain Definite Integrals.” By W. H. L. Russell, 
F.R.S. Received January 10, 1878. 

The integrals in the three preceding papers may nearly all be in¬ 
cluded under the following general theorem:—• 

Let 0(aj)=^ o +^i« + 'W 2 ^ r H- . . . u r x r -f . . . which series is, of course, 
supposed to be convergent, and let II be a general functional symbol, 
such that 

3I0(a j)=»oX( 0), +«ix(l) »+«8X(2) • . • 

and let %(d)=/Ud$, x(l)—/UVdd . . . x( r ) ~j'FV r dO . . . the inte¬ 
grals being supposed to be taken within certain limits: then 

(62.) fF(p(Yx)de=zn<t>(x). 

The following integrals require for the most part other methods to 
determine them: 


(63.) 


dx 


7T 1 f 

= 4 l~~a 2 { 


0 (l J rx 2 ) 2 (l — 2a cos x + a 2 ) 4 

More generally we may obtain 

dx . (65.) [ 

J f 


e-f a 2ae 1 
6— a (e— a) 2 1 


(64) I 

( 66 .) 

( 68 . 

(70.) 


o (l + a? 2 ) r (I ~~2a cosx + a 2 ) 
dx 


sm ax 
o (1 Fx^Y si nbx 

sin ax 


, f 00 dx sin aa 
J o 1 -j- x 2r sin bx 


(67.) [ 
(69.) 


dx . x sin ra 
o (l + £c 2 ) y (l — 2a cos x-j-a 2 ) 

dx cos ax 
J 0 (l+aj a ) r cos bx 

dx , cos ax 
o 1 -j- x 2r cos bx 


(x — a 2 x)dx 


(71.) 


o ((I + a + a 2 ) sin ax—a sin 3<xa?) (I -j- x z ) r ’ 

(1 -a 2 )dx 


o ((1 -\ra+a 2 ) cos ax + a cos 3<xo?)(l + x 2 ) r 
VOL. XXVII. 


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130 


Mr. W. H. L. Russell 


[Feb. 28, 


The integral (68.) calls for some particular remarks. Since by the 
ordinary rules : 

2 sin 0—sin (2r-f-3)0 + sin (2r + l)G 


sin 0 + sin 30 -j~ . . . sin(2r+l)d 
whence also : 


2(1 —cos 20) 


-CO 

dx, si: 
o 


2 sin bx —sin f2w + 3)hr + sin (2n+l)bx 

sm Q/x . - : - 


Now 


/.OO 

1 


(1 + * 2 0(l~c°s 2 bx) 
dx sin ax sin bx ^ f °° dx sin ax sin 3bx 


l + x 2> 


+ . . . +2 


*00 

J 0 


l + x 2r 
dx sin ax sin (2n -f1) bx 


l + x* r 

2 sin ax sin bx —cos (b — a)x — cos (b + a)x 
2 sin ax sin Sbx —cos (3& — n)$—cos (3 b-\-a)x 


f w dx cos mx 

Jo l + x 2r 


= Ci6- c ^+0 2 €~ esm . . . +0,e" c ^ 


Where — Oi, —c 2 , —c 3 . . — c s are all the roots of the equation 
® 2,, + l=:0 which have a negative modulus, and Ci, C 2 , 0 3 . . . Cb are cer¬ 
tain constants whose values have been assigned by Poisson: hence 

f w 7 • 2 sin bx— siu(2w + 3)&aj + sin(2w+l)foe . 

dx . sm ax - - , v ^— = Ci0(ci) + O 2 0(c 2 ^ 

Jo (l + ar)(l —cos 2oaj) 


+ . . . +C*0(c,)i, when 0(c) = (e cx — e^ Ca -) . 


-cb _ 6 -c(2n+3)6 

l_ e -2c—* 


Now let (^) increase without limit, then 

sin(2u + 3) bx =sin 2uj 1 + — )6 oj= sin 2nx. 

\ 2nd 

sin(2n +1) bx =sin 2?i{ 1 -f = sin 2w». 

\ 2n/ 

Hence the integral in the second member vanishes, and 

dx sin ax ^ (e Cia — e - c i a ) 6 ci6 ( 6 c a«— e ~ C2a )e C26 , 

*“U-- 0 ——~— + 0 2 ^- — — -J -—+ ... {s terms). 


0 l + ^ 3? * sin bx 


b ~l 


e^ b ~l 


It is scarcely necessary to observe, that this process implies that (a) 

must be less than (b), otherwise the integral - CQ ^ mX ' as here used 

Jo 1 + ar 

will be discontinuous. 


(7^ ) F dO sin 2? '0_ (2r—l)(2r—3 . . 1 
JoT^+l 2r(2r—2) . . . 2 * 


dO 





1878.] 


on certain Definite Integrals. 


131 


<«■> ]>(»»■<»■ 


Now let us define four quantities, p x , p 2 , /ii, fa, tlius 

v ' 1 +«*+® 4 +( 1 +§)}*’ 


^ + a2+ + 2 ) } 3 


jM'l — Pi Pi f 1? /l2 — />2 + /3l a/3~" 

then we shall have 


(7 6 .)}; 

(76.) 


>a> x 3 -{-a 3 dx __ 

^7~==7r{sin 1 a+sm~ 1 (/j2v'ljH Pi) a }. 


x°—cv 
'* dO cos rO 


l-f-u 3 cos 3 <9 3^1— a? I 


./^i- 


^ 2 -n r + 

(i J 


2tt 


{4 


Pi—r- 


r—1 


2 /h'-V+r • 1 .. -- 3 

P “ 1 r ~Vv 


3(2a) 2 (^+^) 

A»i > 2 '— • .1 


—pi^r^rfii—r. LA. . . .) j . 


(77.) 


(78.) 


(79.) 


n dOe 9 cos 0 


,7r _( 

.oT 


sf i- a a-l 


+ a 3 cos 3 0 3 v 7 !— a 2 


+ 


277* Py{ fa . a \ 

so?+^) e2 “r cos £" /,2Sin £> 

1 dO\og e (l + a? cos 3 6») = 7rlog e 

8 

+ »rloge (1+ v / (« 2 +2v / l + a 8 +a 4 +2) + •v / 'l + a 8 +a 4 ). 

_i r -> r 


r(2a) 


j dO cosrdlog e (l + a 3 cos 8 ^):=-1 

/ P r—1 r _ 2 r , r—1 r 

r. > 2 r + r. — . - 


a 

2 r—3 
4 


i 


+ 




-■) 


(80.) From (79.) we can immediately deduce 

i * m 2 -f-m cos 0 
o 


dO log 6 (1 + a 3 cos 3 0). 


l-f2m cos d+m 3 

(81.) If we denote integral (78.) by jo, and (80.) by q, we have at 


77 dO log e (1 + $ cos 3 d)_ p — 2q 

o 1 + 2m cos 0 -f- m 2 1 — m 2 ’ 


K 2 



132 


Profs. Li vein g and Dewar 


More generally we may obtain 


(82.) 
(84.) [' 


' + 0 ?* 


dx 


x* r+1 4 - a 2, ' +l Vx r —\ 
clO log e (14* a r cos r 0 ), 


and many others. 


[Feb, 28, 


(83> l; 

(85.) 


u cos r0 . dO 


1 ~b cd cos r 0 

d0 log fc - (14 -a r cos r 0) 
14- 2m cos 0 4 ra r 


II, “ On the Reversal of the Lines of Metallic Vapour's.” By 
G. D. Liveing, M.A., Professor of Chemistry, and J. Dewar, 
M.A., F.R.S., Jacksonian Professor, University of Cam¬ 
bridge. No. I. 

Since the celebrated paper by Kirchhoff, “ On the relation between the 
radiating and absorbing powers of different bodies for light and heat,” 
in which he detailed the remarkable experiments of reversing the 
lines of lithium and sodium by sunlight and by the vapours of those 
metals in the flame of a Bunsen’s burner, and mentioned the reversal 
of the brighter lines of potassium, calcium, strontium and barium 
when the deflagration of the chlorates with milk-sugar was used 
instead of the flame of a Bunsen’s burner, further researches in 
the same direction have been made by Cornu, Loclcyer, and Roberts. 
The method adopted by Cornu, which had been previously used 
by Foucault, is one of great ingenuity, dependent upon so arrang¬ 
ing the electric arc that the continuous spectrum of the intensely 
heated poles is examined through an atmosphere of the metallic 
vapours volatilized around them. By this means Cornu succeeded in 
reversing several lines in the spectra of the following metals in addi¬ 
tion to those above-mentioned, viz., thallium, lead, silver, aluminum, 
magnesium, cadmium, zinc, and copper. He observed that in general 
the reversal began with the least refrangible of a group of lines, and 
gradually extended to the more refrangible lines of the group; and he 
drew the conclasion that a very thin layer of vapour was sufficient for 
the reversal. It may be noted that in almosty every case the lines seen 
by him to be reversed were the more highly refrangible of the lines 
characteristic of each metal, confirming generally the opinion expressed 
by Stokes in a letter to Lockyer in the Proceedings of the Royal 
Society for 1876, in which he. introduces for the first time the idea of 
the persistency of different rays with reference to temperature. 

The method adopted by Lockyer in the first instance was to view 
the electric arc through the vapours of the metals volatilized in a 
stream of hydrogen in a horizontal iron tube. 




