14 Mr Oreenhill, Complex Multiplication [Oct. 29, 



This has been verified arithmetically by Mr Pilkington, and his 

 verification is here appended. 



If c = (7 + 4^3) (4 -^15), 



then jTc=(2 + 73) (J 5- J 3) ; 



and if 



1 — y _ 1 1 — x ic+x /J '— ic 4- x\ 2 

 y-ic Jjc" 1 + x ' ic - x \J- i c - aJ 



_iji ic — x 2 +(l—ic)x ic — x~ — 2xj — ic 

 ~ Jc ' ic - o? - (1 - ic) x' i c -x 2 + 2x J^Tc 



_ i — 1 (ic — x 2 ) 2 — 2x 2 J —ic(l— ic) + (ic - x 2 ) (1 — J — ief x 

 ~ J 2c ' (ic- x 2 ) 2 - 2x 2 J^lc (1 - ic) -(ic - x 2 ) (1 - J^icf x ' 

 then 1 — y 



_ ( j-l + c + ic) {(ic — x 2 ) 2 — 2x 2 J- ic (1 -ic) + (ic-x 2 )(l-J -icfx] 

 = _ 



where D = (i - 1 + J 2c) {(ic - xj -2x 2 J^Tc(l- ic)} 



. .+ (t - I - J 2c) (ic -x 2 ) (1 - J^icf x } 

 and 1 + y — 



(i-l + 2 J2c ~c- ic) { (ic - a: 2 ) 2 - 2a: 9 J^icjl -ic)} + (i-l-2 sj2c- c—ic) (ic - a 2 ) (1 - J^icfa 



D 



This should be of the form 



P{a 2 /3 2 +2a^(a+ i 3)a ! +[(a+/3) 2 +2^]^+2(a+/3)a ; 3 +a ; 4 } 



whence, by comparing, we must have the following relations 



P = i-l + 2j2c-c-ic (1), 



a 2 /3 2 = i 2 c 2 (2), 



(a+/3) 2 + 2a/3 = - 2ic - 2 J^w (1 - ic) ... (3), 



a or i o\ i-^-^J^c-c-ic . . I — - N , ... 



2j/3 a + /3) = . /- r .ic(l-7-.tc)- (4), 



t — 1 + ^x/ZC — c— IC 



