On the Analysis of Square Numbers. 89 
2. If A?7=B?+C?, then A*=D?+E?=F*+4G?; that is, if 
' any square number contain the sum of two squares, the square of 
that number will contain the sum of two squares in two different 
ways. 
Demonstration «4 =(b? +c?)? =(b2 —c?)? +-(2be)? =076?-a%c?. 
3. If A‘ and B* contain each, the sum of two squares in two 
different ways, then A‘: B‘ will contain the sum of two squares in 
twelve different ways. 
Demonstration. Put A*=a? +b? =c?+d? 
and B4 =m? +n? =y? +2? ; 
then A‘: Bt at +n? )*(a2 4+-53)=a? 46? =v? +62 
a ip a(n? +-n?)-(c?+d?)= 2 +23 =? +42 
ee =(y* +27) (a? +5?) =; +x? =Dr?2 +, 
=(y? +27)(c? +d?) =v? +22 = 0? 4 #? 5 making in 
all eight en of squares, deducible from the general expressions [A] 
and [B]. In addition to these, we may derive four sets more from 
the simple multiplication of the expressions representing the value 
of A‘ 
For (m? +n): (a? +52)=m?2(a?+5b?)-+n?(a? +b? )=m?A4 intAt, 
(m? +-n*)*(a? +52) =a? (m?+tn?) + (m?.4n?)=a? Bs + b?B4, 
(y? +23 y (c? +d?) y=y s(e*. +d?)+27(c? +d?)=y?A‘tz7A4, 
(y2 be? (e202) 08 (y2 +29) +d2(y? $24) 0B! 4B. 
: Q. E. D. 
Remark.—The product A*-B*=(A’B)*. It may be shown that 
A and B each equal the sum of two squares when A? and B? each 
equal the sum of two squares. ‘Therefore the fourth power of the 
product of any two numbers that are prime to each other, and that 
consist each of the sum of two squares, may be resolved into the 
sum of two squares in twelve different ways. The smallest number 
that can be thus resolved is (5.13)*=65*. In the same way it may 
be shown that the expression (A- B)°, or the number 65* may be 
resolved into the sum of two squares in eighty four different ways. 
4. A convenient method for finding two Janae whose sum shall 
be a square, is the following. Let a?— For ¢? put any 
square number whatever; then, by the common es representing by 
m and n, any unequal factors of c*, we have os", and b= "> 
Putting for c? any square pumber a?€2~% where a,S,y, represent 
prime factors, we have 2a—=a?6? piblmattry tymetttty 
Vor. XXV.—No. 1. 12 
