050 



PROFESSOR CHRYSTAL ON THE HESSIAN. 



The diagrams here are both of the same character ; 

 that for U, for example, is figure 3, where AB and 

 CD are parallel and each full of terms ; the co-ordinates 

 of D are (m , k- m) ; and O A = k -f- r , OC = k . The two 

 lines CD and BD give approxmations at x = y = 0. 



We may therefore write 



U = u k - m (x m + y r + m ), 



V^v^i^+y^) 



Hence 



according as 



UY = (k-m)K + m(K-p.) + l{(p + /ui.)m,(r + m)p.} 



= &/c — nifx + mp. + l(pm , rp) 



= hit + l(pm , rp.) 

 i.e., = kic + pm or = kK + r/m, 

 pm < or > r/j. . 



This result includes that of Ex. 2 in § 2 as a particular case. 



Ex. 4. To illustrate the particular case of Ex. 3, where mp<^r, consider 



Here m = 2, &=2, r = 2, 



,01 = 2, /c = 2, p = l ; 

 mp = 2 , p.r = 4:. 



UV = 2x2 + 2 = G. 



The corresponding figure is 



which may, in fact, be considered as the limiting case of 



(To be continued in another Communication.) 



