359 


1877.] On certain Definite Integrals . 

about 13' nearer tbe normal of the face (10 0) than that of the blue, the 
mean direction of the first mean line being about 35° on the normal of 
10 0, and 8° 40' on that of 101. 

Crystals of potassium nitrate, containing less silver nitrate than the 
double salt, gave angles according somewhat closely with those of ordi¬ 
nary nitre ; but the crystals did not give very good reflections. 

The results obtained by Rose with sodium nitrate containing silver 
nitrate were confirmed so far as the crystals permitted of measurement. 

Crystals of strontium nitrate containing silver nitrate gave excellent 
measurements, according with those of a cubo-octahedron. 


VIII. “ On certain Definite Integrals.” By W. H. L. Russell, 
F.R.S. Received June 21, 1877. 

The following paper is a continuation of two papers recently commu¬ 
nicated to the Royal Society, and inserted in the 4 Proceedings.’ 


J 7T 

dd log e (e 2 * cos e + 2e x cos 6 cos (x sin 0)-f l)=27r log e 2. 
o 

J IT 

2 dd e 2 008 6 cos (sin 20 cos 2 0 -f sin 0 cos 0) = ^ i / € 3 . 
o 

J 1 IT 

dd e 008 20 + CO9 0 cos (sin 20 -f sin 0)=7r. 
o 

Pdfle 1 " 

J o 


(43.) 


^ oofecos . 2XSin9 . 


1 — 2x cos 0+ x 2 


C * 1 ~ x 0090 x sin 0 

(45.) \ 6dQ.- 

v 0 


sin 0 (4+cos 2 0) 7r 

(3 + sin 2 0) 2 3 


sin 3 0c?0 7r(^ 2 —1) 1+at rr 

----log e —-4- —• 

x 2 cos 2 0 2x 3 6 1 —x * x 



0 sin 3 0 dd 
l+x z cos 2 0 


^ 2 +l) 

X s 


tan ~ l x — 


7r 



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360 


Mr, W. H. L. Russell on 


[June 21 


Continuing the process, we find 
f" 0sin 5 0<70 

(48 ° J o l+i T eos 2 e’ and ” generally (49 


•>££ 


sin 2w+1 Odd 


j: 


dO sin 2 a0 


1 4 x 2 cos 2 0 
7T € 2a — 1 e a 


((1 + ju+ju 2 ) cos ad+ iu cos 3 ad)) (1+ 0 2 ) 2-2^'e 2a +l‘e 2a +/ 


(51.) I a clQ e cos n ° cosn0 cos (cos M 0 sin nd) = ^/ e . 

h 2 


(52.) (*"^sm|e^ 8 sin(0+|) 

^ 0 1 — 2a cos 0 4 a 2 ~ 

TT 

0 


7 r e® 4 

2f+a' 


(53.) 




COS 0 + c 

d6 (1 + e 008 0 cos sin 0) 


(1 —2a cos 0 + a 2 )(l + 2jLie cos0 cossin0+/i 2 e 2c080 ) 1 —a 2 l + fte a 


dO (1 + fie 003 6 cos sin 0) 


0 (l + 0 2 )(l + 2/x€ cos0 cossin0+^ 2 e 2cos0 ) 2 p_|_^ € 


J > 7T 


(55 
(56.) J 


e co&e sin (sin 0) <70 sin 0 


2a cos 0 + a 2 ) (1 + 2jue cos0 cos sin 0 +fi e 2 
log e (1 — 2a cos 04a 2 ) 2?r 


2a } 1 4 jue a 1 +fi J 


l + 2.rcos 04# 2 


l-*r 


. log e (1 -}- Otf). 


In like, manner we may find the integral (57.) 


•>f 

n 


<70 


(14 0 4 ) (1 — 2a cos 0 4 a 2 ) 9 


J 00 <70 

7“- r ~ — — -“— 77 . 

(14 6 2 ") (1-2a cos 0 4 <) 

f -_ <70 . sin 2 0 ___ 7 r 1 

’ J 0 (1 — 2a cos 0 4 a 2 ) (14 2a? cos 0 4 a* 2 ) 2 ~~ 2(1 — x 2 ) ’ (1 4 a#) 2 * 

J °° 0 sin 0 <70 7 re 

0 (1 4 0 2 ) ( 14 2# cos 0 4 a? 2 ) 2 2(1 — x 2 ) * ( e 4 #) 2 * 

f dO mud 7t (e— 1) (e — x 2 ) 

(6 ^ J 0 (040 3 )(142^^804^7^ 2a-^)(l4^ 3 (€4^ 9 



1877.] 


certain Definite Integrals . 


861 


I hay© been asked to indicate the method of obtaining some of these 
integrals. This I now do. 

(8.) is obtained bj summing the series for by means of the defi¬ 


nite integral 


£ 


log e cos 0 cos 2rd = ( — iy-1 

4 r 


:>A+" 

by means of the definite integral 


(9.) by expanding V l + ~in terms of a, adding the resulting series 
2 


£ 


2 clQ cos n 0 cos nd = ~ — 


2 2 


n ’ 


and combining the definite integrals arising from the process, by the rule 
for the addition of binomial surds. To obtain (10.) we expand —— 

in terms of a, and then sum the series by means of the integral 


£ 


, 7r 1 

' dd cos n 6 cos nO = -. 

2 2 n 


(18.) is found in a similar manner. (14.) is obtained by using the integral 


£ 


w dO cos a0 


1 + 0 2 


, and (16.) by means of the integral which we used to ob¬ 


tain (8). (19.) is found by summing up the sexles for 


C* 

of the definite integral J 6 sinn0=(—l) w+1 -. (40.) is derived from 

(19.) by integrating by parts ; but this integral can also be obtained by 
expansion. (21.) and (22.) are obtained from the definite integral 


£ 


dx 

€*+1 


by means 


£ 


2 cos n ~~ l d dd cos {c tan 0+(n —1)0} d0~-~€- c . 

2 


(33.) is found by using the integral 


£ 


dO Q cos 71-1 0 sin (n-f 1) 0 = 


2»+i *n 



On certain Definite Integrals . 


[June 21, 


(34.), (35.), and (50.) arise from the integrals 


cos 2rad 


I sin ad (1 -f 6‘ 


. Odd C” 


cos 2 rad . dd 
cos ad (1 + d 2 ) 2 


(53.), (54.), and (55.) are obtained by using series analogous to 

€ cob o cos s i n j U€ 2 cos 0 cos (2 sin 0)4-jli 2 e 3 eos 0 cos (3 sin0)4- ..., 
and (59.), (60.), and (61.) are found by the series 

off sin 0 — 2oo 2 sin 20 4- 3#® sin 30 —. . .=— 


( 14 - 2x cos 0 4- off 2 ) 2 * 


The series employed are considered as convergent, according to the 
rule given by Cauchy. On account of the great importance of the sub¬ 
ject of convergent series, I have thought that mathematicians would be 
interested to see how the results of Cauchy are confirmed by a method 
given by me in the 4 Proceedings of the Royal Society’ for 1872, vol. xxi. 
page 20, and which was afterwards discovered independently by Sir W, 
Thomson. Let u^+^oo 4 - u 2 a? 4- ... 4 - u n oc n + ... be any series, then 

I call when n increases without limit the ultimate ratio of the 
u n 

series, and denote it by p . Then if 00 is less than the series will be 
convergent* 

Now consider the function which we have employed in these 

€ i 

investigations. 

Since 77-7 4- ~Z“TT —L the series comprising —— can contain no 
6*4-1 c * 4~ t 0 6*4-1 

even powers, and therefore we may assume 


j-==?.* 0 4- ^off 4- V 3 + V 5 4- . . . 

Expanding 6*4-1 in terms of 00 , and multiplying the series together, 
we obtain 

o , w 2rc~l , U2n- 3 

2 M 2« + 1 + 172 "+ 772 . 3.4 + ••• =0; 

orl + l+!^^A +—•—+ ...=0. 

tl2n- 1-1 1 * " < ^ > 2n —1 U2n- f-1 1.2.3.4, 

Let n increase without limit, and then 

n,2n —1 

__ • . • » ^ p, 

n-2n—l 3 



Presents. 


363 


1877.] 


I 

whence 1 -f- cos -j s 

pi 


, which gives 
1 


.p- or 9jr „ or 25jr , • 


that is to say, the series will be convergent if —, ^ ♦ . • are all 

7T 97T 207T 

less than unity—that is, if x is less than 7 r; which accords with the 
result given by Cauchy. 

Let us next consider the series 


e z ‘ i+x = u Q + u x x + u 2 x 2 -f . . ., 

we easily obtain by differentiation the following equation for deter¬ 
mining u M 

(n + 1) u n+ i = u n + 2u n _ h 


which may be written 


O+i) 


^n+ 1 Myi 


Un 


u n —1 


= -=-+2. 


» n~i 


'll xi U 

As n increases without limit, = —— = p, and the equation becomes 

u n—\ 

p 2 ass 0, or p =* 0, whence the series is convergent however great x may 
become. 


Presents received, June 14, 1877. 


Transactions. 

Batavia :—Bataviaasch Genootschap van Kunsten en Wetenscbappen. 
Tijdschrift voor Indische Taal-, Land-, en Volkenkunde. Deel 
XXIII. Afl. 6 & 6 ; Deel XXIV. Afl. 1-3. 8vo. 1876. Xotulen 
van de Algemeene en Bestuurs-Vergaderingen. Deel XIY. Xo. 
2-4. 8vo. 1876-77. Catalogus de Ethnologische Afdeeling van 
het Museum. Tweede Druk. 8vo. 1877. Yerslag van eene Yer- 
zameling Maleische, Arabische, Javaansche en andere Hand- 
schriften. 8vo. 1877. Het Maleisch der Molukken. 4to. 1876. 

The Society. 

Boston [U.S.] :—Bussey Institution. Bulletin. Yol.II. Parti. 8vo. 

1877. Harvard University, 

CherbourgSociete Xationale des Sciences Xaturelles. Compte- 
Eendu, 30 Dec. 1876. 8vo. 1877. The Society. 

Dublin :—University Biological Association. Proceedings. Yol. I. 
Xo. 2. 8vo. 1876. The Association. 



