395 

 Or, replacing OB in each case by BO, we have 



f(x ^ -X' M i P M P v m 'i P m * (m) 

 'BO'" BO * W 1BO 



where, since there are w + 2m factors, 



6. But if, instead of simply changing the origin, we change the 

 inclination of PQ and make it parallel to OC, then if OB and OC 

 make the angles /3 and y with OI, and O J be at right angles to OI, 

 the lengths of OI, O J, OB, OC being the same, and M be any point, 

 we have 



and 



OM=#'.OI+y'.OC=(*'+y f cosy). OI+y' sin y.OJ, 



which give two simple equations to determine x, y in terms of #', y' t 

 so that on substitution f(x, y} becomes $(#', y f ), a formation of the 

 same number of dimensions. But the coefficient of y' n +* m will be 

 different from that of y n + 2m . Let it =/u. The curve for the scalar 

 values of x\ y' in 



oM=*'.oi+y.oc, 0<y,y)=o 



will be precisely the same as that in (3) ; but as PQ, having a differ- 

 ent inclination, will cut it in very different places or not at all, the 

 numbers of scalar and clinant roots of 0=0 may be individually dif- 

 ferent from, although their combined number will be the same as, 

 those of/= 0. If then there be h scalar and 2k clinant roots of 0= 0, 

 where A-f 2k=n + 2m, and we determine N t , N,...^, N' 2 . . . in 

 the same way as we previously determined M x , M 2 , . . . M' I} M' 2 , . . . 

 we find 



N.Q N A Q N\Q N a WQ 



/(*, *)=** *)=/ ^- w x oc ----- fee- 



7. This investigation shows that if there be any number of lines 

 drawn from a point, cutting those radical loci of an equation for scalar 

 and clinant roots which correspond to the inclination of these lines to 

 the axis and to the position of the origin, coefficients X, /u, &c. can 

 always be assigned such as to make the product of these ratios each 

 equal to the other ; and that these coefficients have only to be multi- 

 plied by ( l) w+2m , in order to give the coefficients corresponding to 



2F2 



