230 EEV. T. P. KIEOIAN ON THE K-PAETITIONS OP THE E-GON AND E-ACE. 
^ . Sr + w Zh—2n 
2n 
* T^/o 1 s + 3A— 2n\ 
A7>-2a=D(2-f«i, ^i)— D^' 2„--5 2^ J’ 
and this is 2 A^, as there is no other solution of (A.). Wherefore 
'Sr + ra Sk—2n^ 
ih 
Ef*^, *).=d(^ 
2ra ’ 2n 
Next let ^-r^=2, or 2A=^; we obtain 
4/^ 
4A 
which is a given number. Here, as before, 2A has as many terms as there are solu- 
tions of 
r=?z-fy(«,+«2) 
, 4ra 2ra, , , 
k— .y + -j(ei + e2). 
71 
For an example, let ?’=78, ^=48, 7^=30; ^= 6 : to find R®'^'(78, 48 ) 30 , form the 
solutions of 
4=«,-f-«2 
2=^1 + 62 ? 
of which there are five, namely, 
4=4+0= 0+4=3+l = l+3=2+2 
2 = 3-l = -l+3=2+0=0+2=l+l; 
whence, disregarding the negative — 1 , we obtain 
2.D(6, 3)+2.D(5, 2).D(3, 0)+D(4, 1).D(4, 1)=SA„_:^, 
or 2A„^=2. 14+2.5+2^=42; 
4h 
and 
because — is not a whole number. 
4h 
J^a.a^d*(78, 48)30=42 -R'«“^'''(78, 48)30=42, 
20 
30-2/j 
We may then put = 7, the next possible integral value after 2 of that sub index; 
and thus finding h=l, proceed to enumerate Rf^*'(78, 48 ) 33 . The equations (A.) become 
78-30 
4 
48-28 
— 12 . . +fl/- 
4 ■ — ^ + 
and the solutions are already numerous. The sum of products A- being found, we obtain 
R2.a^rf^(78, 48)3 o=2A.-42. 
