183] ST. TENANT'S PEOBLEM. 481 



Comparing with 81), we find that both of these derivatives must be 

 zero. Accordingly we have 



Q ON d 2 (dw\ d 2 fdw\ d 2 (dw\ a 2 /dw\ A 



dF* \d~zj - df \R) = W (w = d^rty VW = 



Hence - cannot contain any power of x, y or above the first, 

 nor #?/, so that we may put 



84) ^**<* 

 Hence from a), 



r\ r\ 



85 ) Jx == Jy = ~ 



Differentiating 84), equations c) and d) become 



Accordingly w and t; contain powers of not above # 3 , M contains 

 powers of x not above ^ 2 , v contains powers of y not above y 2 and w, 

 powers of z not above # 2 . 

 Integrating 85) and 86), 



1 



87) 



= 



-f 



Putting these in b), 

 - ^ ( 



- ^ Ky + &i^) + ^y' W + /"' 0*0 = o, 



so that ty, %j tpj f Sire of the second degree, in the variables indicated, say r 

 Z(y) = a' -f < y + a 2 f t/ 2 , 



in which 



< = 

 90) 



v r ~ 



, Dynamics. 



