88 On the Analysis of Square Numhers. 



Remarks. I. When neither a and h, nor c and d, are equal to 

 each other, it is evident that the product A-B contains the sura of 

 two squares in two different ways. 



II. When a=h, both powers are reduced to one; viz. 



K'B = {ac-\-adY-\-[ad-acY, 



III. When i = 0, both forms are reduced to one ; viz. 



K'B = {acY-\-{adY, 



IV. When a =c, we have K'B^{a^- -\-hdY-\-iad-ahY. 



and A.'B = la^ —hdY +\ad-\-abY ', 

 expressions which still afford two different sets of squares. 



V. If a=^c, and& = c?, the expression [A] becomes {a^-\-h^Y — 

 [a^ +b^Y + {ad-adY ={a^- -^h^^Y ■{■{OY ', while [B] assumes 

 the form, given by Souri, Bonnycastle, and others, for finding two 

 square numbers, whose sum shall be a square: viz. {a'^-\-h^Y=- 

 {a^ -h^Y +{^ahY ' 



VI. If a=h, and c=d, both expressions assume an identical form, 

 (2ac)2=(2ffc)% from which nothing can be determined. 



VII. If a = \, and 6 = 1, both expressions [A] and [B] take this 

 form, 2(c2-fj2) = (c4-</)2-|-(c-£?)2. Whence it appears that if a 

 given number contain the sum of two squares, the double of that 

 number will also contain the sum of two squares, but generally only 

 in oneway. 



VIII. Finally, if a^ +h^ =0"^ -{-d^ , while neither the values of a 

 and J, nor of c and d, are alike, we have the expressions, 



A-B=A2 = (a2+62)2 = (ac+W)2 + (a^-Jc)% and 



(«^+62)2=:(ac-5rf)2+(a(^-f6c)2; that is, a 

 square number that contains the sum of two squares in two different 

 ways. 



Kote. — The condition a^ -\-b^ = c'^ -\-d^, is satisfied at once by 

 substituting the forms of A*B, exhibited in [A] and [B]. But if 

 a^-\-b^=c^-^d^ ; then {a^ -{-b^yim'' -\-n'') = {c''+d^y{m''-\-n''). 

 Now (a" + b^y{m^ -]-n^)={amtbnY +(an:^bmY =a2 +§2 =y2_|.^2 

 and [c- +d'y{m^ -{-n'^) — {cmtdnY + {cn^dmY =s^ +<^^ ='^^ -{-&^ , 

 by employing the change of signs^ But 



(a==+§2}:(72 4-52)=(a2^g2)=^(a7+g(5)==+(a,5:;ey)2,and 

 (j2 ^^2).(^2 ^^3)=,(„2 +g)2 = ( Byit^dY-^i'^+^-^y- Thus we 

 have found a square number, (a'^+Q^)^, which may be resolved into 

 the sum of two squares in four different ways. In this way we may 

 proceed, till we have found a square number that can be resolved 

 into as many sets of squares as we please. But a better method is 

 the following. 



