470 


Mr. W. H. L. Russell on 


[Jan. 25, 


January 25, 1883. 

THE PRESIDENT in the Chair. 

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

The following Paper was read:— 


I. “On certain Definite Integrals.” Ho. 11. By W. H. L. 
Russell, F.R.S. Received January 16, 1883. 

The method by which integral (227) was obtained may be thus 
extended. 

Suppose it was required to obtain 

J \a' + b'x-\-cx 2 -\-e'x 2j r . ../ 

Put z= a + bx+Qx 2 -\-ex 3 + ■ ■. 

a f -\-b'x+c'x 2 + e f x 3 -\- . . J 


0(x)=za+bx+cx 2 +ex 3 + .. 

0(a?)= a f 4 Vx 4* cfx 2 4 e'x 3 4 ...., 

then 0(a?)=20(a?), and if (<*) be any root of the equation O(x)=0, and 
6 . . . 0', 0", . . ., are the values of O'x , 0 f, x when x~a, then we 
find by differentiating 0 (#)= 20 (aj) that 


x=TJ 0 +TJ 1 + -U^+-U sr ^-+ ... 

vhen U 0=ai) U 1= |„ U 3 =^-^, 

TT _3f/>"0 2 90W' , 60<Z>' 3 ?j0"cjy’’ 

Us ~ e's e'4 + o's e'i ^ e'5 • 

Hence f^/(-^ a,+c ^± ea >+—) 

J J \a 4 V x + cx 2 4 e x 3 4 .*./ 

=j&(u 1 +U ja +F 3 . i ^+...)/W . (234). 

The theorem applies with great facility when f(z )=\/I of course 


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1883.] 


certain Definite Integrals . 


471 


suppose that f(z) can be expanded in terms of % by a converging 
series in other cases. The series U 1 + IJ r 3 z+ . ..would, I believe, be 
convergent in many cases, where O' is large when compared with 
0, 0', ...; bnt the subject requires further consideration. 

If We put 25=0, 


Hence we immediately deduce from the equation 
*=V 0 +U 1 8+TJ 2 j?-_ + .. 

the theorem . 


(235) 


=0.. 


Since 


P f dxx n ~ l = -1 , 

JoJ n(n+ 1) 


f l r 

d 

Jo. 




^dx(l + x)x n ~~ 1 = 

J^23(l— x)x n ~ l = 
and, therefore, as before 
j" 1—23) 

_ A 0 _J_ ^1 *1*2 

- 7- rrv + 7 .tT7- +- 


1.2 


w(w-f-l) (w + 2)’ 

_1 .2,3 

n{n +1) (n + 2) (n + 3) ? 


A 2 .1.2.3 


- + 


(236)} 


n(n-\-Y) n{n-\-V}{n-\-2) n(w + l)(^-f-2)(^+3) 

with precisely similar formulas for j* JJ dx^x n ~ l (fi(Y—x) and other like 


integrals. 

We also have 


1 0 ( 1 — x^dxa$ n 

o — x* 


i 


-f A i + ^•1-3 


2^ + 2 (2^+2)(2?i+4) 

dx 


+ 


n \ f 1 dxx* n 

’MJ.TTSS 


(237), 


(l + x 2 ) n 

= { Ao+ fe + 2^Tl) +!&c -H 


dx 


o (l-j-a; 2 ) w 


. . (238); 


It is easily seen that these investigations extend to a great multi- 

2 i 2 



472 


On certain Definite Integrals . 


[Jan. 25, 


tude of what are usually called binomial integrals. It will be seen 
that these formulae include a vast number of such integrals as 


dx . x 2n 

. v 7 cfi — x 2/ s/l— a 


(239). 


da j . 


\y/a 3 —a ? 3 1 —a ? 3 


1 

fc 


dx . £C 3w 


V 7 a+&a2 2 + ca3 4 + • • • +eaj 3w ^/l—a? 3 
dx . ® 3w 


/ V^a-4- coj 4 4- . -. 4- ex 2 * 1 */1 —a? 3 
If we have an integral of the form 

/d0€ P *Q = 6^, 


(240) . 

(241) . 

(242) . 


Jl 


where P and Q are functions of 0, we have by applying the symbol 

d. 

0— to the integral 

dx 

/id0(P)e p ^Q=0(a)a^ ..(243). 

As an example of this 

jrdd{0(cos 3 de 3 ^) 6* C0S3 + (0 COS 3 6e~ m )e* c09 ^~ * id }=* jr0(|) e* (244), 

Jo 

from whence 

£ e coss a cos (® COS 3 d sin 3d) cos 3 0 cos (x cos 3 6 sin 3d—3d) 

1 + 2/4 cos 3 d cos 3d+ fi 2 cos 6 d 

=^8" • • • ^ 

In the same way we may deduce general formulae from integrals 
(86), (87), (116), (118), (122), (129), (130), of the present series. 

The following integrals were obtained by the use of reciprocal 
functions:—- 

j* cZd cos r de acos 0 cos (a sin d+rd)=. . . . (246), 

. . (247), 
w(l + a s (3 r J 


J: 


do cos' 0 . cos^-acos_(r = l)0 = _7 L 
1—2*cos0+* s 2 r 




d0 


- 2 a cos rO + a' 2 ) (1 — 2/3 cos sO + ft 3 ) (1 — « 3 ) (1 — /3 3 ) (1 — a 8 /}') 

. . . . . . (248), 

where r and s are prime numbers. 

|’ r d6'{6“' 9i 0(cos0£«) + 6' tc ~ e, '0(cos £>£-«)}=2 jt0(|) . . (249). 




473 


1883 .] Internal Reflexions in the Eye . 

This last formula is derived from (246). I observe that the funda¬ 
mental idea by which these integrals are obtained is given by the 
equation 

( (A 0 + A-.COS0-K . A,.cosr0)(B r cosr0 + B r+1 cos(r + l)0 + . t )=^A r Bf. 

J o 2 

This method may be much extended. 


II. “ Internal Reflexions in the Eye.” By H. Frank New all, 
B.A. Communicated by Dr. M. Foster, Sec. R.S. 
Received January 18, 1883. 

1. The observation I have to record first came under my notice 
three or four years ago. Often when working at night by the light 
of a candle, in a room otherwise dark, my attention was caught by a 
very faint light some way out of the line of direct vision. This 
seemed to defy nearer inspection; for the instant I turned my eyes 
towards it, it was gone, thus showing that there was no objective 
cause, but that the light was due to some internal reflexion in the 
eye. Later, however, I found that by keeping the eye fixed and 
moving the candle, the faint light could be observed at leisure, 
though, as far as I could then make out, never- in the line of direct 
vision. (See however below, § 32.) 

2. The best conditions soon became apparent, and I have applied 
two methods in later investigations : (i) One in which the eye is 
fixed on a spot on a dark or uniform ground whilst the candle is 
moved to and fro out of the line of direct vision, (ii) One in which 
the candle or source of light is kept fixed, whilst the eye follows the 
regular movement of some point, such as the end of a pencil moved 
by the hand. 

3. The first of these methods showed that the ghost, as I may call 
the faint light, moved roughly speaking in a line drawn through the 
point of clearest vision and the candle, in direction opposed to that of 
the candle’s motion with respect to the point of clearest vision, and 
with a velocity equal to that of the candle. 

4. The second method showed what is practically the same thing, 
namely, that the line of movement of the ghost was just as described; 
the direction the same as that of the point of clearest vision over the 
field in front; the velocity apparently about double that of the point 
of regard. 

5. In both methods the ghost merged into the candle close to the 
point of direct vision, and in other positions seemed about equally 
removed from that point with the candle. 



