Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 1)

2] 
SYMBOLICAL EXPRESSION. 
II 
Or the function <f>(a — x, b — y...) not becoming infinite within the limits of the 
integration, we have 
ff... (f) (a — x, b — y ...)dxdy ... 
‘‘¿Ilk, ... 7T*” 
£ 
r (|n) ^°2^ +1 .1.2..._p4n(|n + lj...(Jn+jD)r da- 
the integral on the first side of the equation extending to all real values of x, y„ 
h^h- 
1 
0C~ f lï~‘ 
&c., subject to -t^ + To+ ••• < 1. 
Suppose in the first place <6 (a, b ..0 = - 
F V (a 2 + b 2 ...f n 
By a preceding formula the second side of the equation reduces itself to 
2hh,... 7r in f 1 a?” -1 dx 
y being given by 
Hence the formula 
r (in) J o V{{rf + h 2 x 2 ) {if + h 2 x 2 ) ... (n factors)} ’ 
a 2 6 2 
if + h 2 rf + hf ’ “ 
dxdy... 
... n times 
2hh,... 7r in f 1 
{(a-tf) 2 +(£-y) 2 ... } in 
¿c n 1 . dx 
r (in) Jo VK 7 ?' 2 + fdx-) {rf + hfx 2 ) ... (n factors)} ’ 
where the integral on the first side of the equation extends to all real values of 
x, y, &c. satisfying j- +y- + &c. ... < 1; if, as we have seen, is determined by 
a 2 b- , 0 
^+^ + ^+^ 2+&C - -1; 
and finally, the condition of <p{a — x, b—y...) not becoming infinite within the limits 
a- b' 
~hr- + h: 
of the integration, reduces itself to ~ 4-... > 1, which must be satisfied by these 
quantities. 
Suppose in the next place that the function </> (a, b...) satisfies + &c. = 0. 
f d 2 \ 
The factor f/¿ 2 &c.J may be written under the form 
+<*«■ ~ ^ +&c - + » (i + m ■■)=<*• - ,i!) w +&c - 
2—2
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.