126 Sir "W. Thomson on Cauchy's caul Green's 



fiy, we find 



^zA- = ArB_ ran- 



whence (sum of numerators divided by sum of denominators), 

 B-C = C-A = A-B 



<ya — a/3 a.(3 — fiy (3y — ya. ^ '* 



The first of these equations is equivalent to the first of (37) ; 

 and thus we see that the two equations (37) are equivalent to 

 one only ; and (39) is a convenient form of this one. By it, 

 as put symmetrically in (40), and by bringing (14) into 

 account, we find, with k taken to denote a coefficient which 

 may be any function of (a, /3, y) : 



A = *(S-/3 7 ); B = *(S-y«); C=*(S-*£);*> 



where S=K0Y + V* + "P) ■* ' 



and using this result in (23), we find 



L=A[ a (/3 + 7 )-S]; M = *[/% + a )-S]; N=%(«+/8)-S]*) 



or L = £(2S-/3 7 ); M=£(2S-ya); N=/c(2S-«/3) J ( ^ 



By (2) we may put (41) and (42) into forms more convenient 

 for some purposes as follows : — 



A-*(8-£); B-J(B-J); 0-*(B-i) . (43), 



L=i(a8-i); M = A(2S-i): !-*(«- 1). (41), 



where 



8 -*(; + W) < 45) - 



Next, we find G, H, I ; by (38), (44), and (45) we have 



G + H + I = KL + M + N)=l^S=^^-f|+ 1 ) . (4(5). 

 whence, by (38) and (44), 



G-ft(i-4B) s &-*(J-ts); I-*g-ifl).(47). 



20. Using (43) and (47) in (19), Ave have 



V ^ V _ , s /«« 8 , 8/3 2 , S 7 2 \ ( (4 ° 



