194 Sir James Cockle on the 



which reduces to 



(2(0 + 3) {©(© + 3) + 3b\ =b 2 \/9^4~b 2 ; 

 then, since 



(2a> + 3) 2 =4a>(<B + 3) + 9, 



if we introduce an auxiliary W, such that 



W = a>(a> + 3), 

 the above equivalent for (1) 2 can be put under the form 



(W + 3&)v / 4W + 9 = (b - W)\/9-46 + 4W; 

 while the last equation of art. 23 becomes 



W + 35=-2a>\/9-4& + 4W, 

 if we transpose. Hence, by division, 



V'4W + 9( = 2e0 + 3)=-^r; 



or, substituting for W and reducing, 



3©(o) + l)=— 6, 

 whence 



25. But, squaring the equivalent for (1) 2 , 



(W + 36) 2 (4W + 9) = (6- W) 2 (9 -46 + 4W), 

 whence 



(3W + 6 + 3) 2 =9-126, 

 and 



w=©(©+a)=-i-|+- v /i=p. 



26. Solving this quadratic in o>, 



or 



<»= _J(5+Vl^p); 



of which solutions the latter is irrelevant, because it does not 

 solve the equivalent of (2) 2 . But the former satisfies both (1) 2 

 and (2) 2 . 



27. Recurring to art. 22, and writing (7) in the form 



6(N + L) = l-(&-l) 2 -a> 2 , 

 and eliminating i-l — 1 between (6) and (7), we find 



l-6(N + L)=(i) 2 (N-L)^+< B ' ; . . (8) 



