MINDING’S SYSTEM OF FORCES, 521 
This arranged according to powers of A, gives 
\(g—h)2y222 
—MA(g—h) (hy? +92 + (9 +hy(e+ gu +h)j yx 
+72 { (hy? + 927)? + 2(9? —?) (hy? — 92") — (9g —h)’y?2? + 2(9° — h?)aP(2? —y")} 
+ 4(g—h) thy? +92 + (g+h\(e—-g)(#—h)s yz 
+0°(g—hyy?x? 
= On” , ; , ‘ (7) 
Thus through a given point there pass in general four lines of the system. 
The lines of single resultant, therefore, form a congruency of the fourth order. 
It will be seen, however, that 
2 a2 
either 2=0and B— go = aE a es : ‘ ; ; (8) 
ae 2 
or een aga a | ; ‘ : . : (9) 
makes the biquadratic (7) indeterminate inasmuch as either of these assumptions 
causes all the coefficients to vanish. 
Hence we conclude that every line of the congruency intersects both these 
conics. 
This may be farther verified by observing the values of Ay» given by (5) 
and (6). When we put z=0 we get alternatively 
Ma OR Ree em eo oy (00) 
(g+h)(@—h) | 
or Ska a 
Eo ecg eID 8 ave wap vi (11) 
iA hy | 
(11) in conjunction with (4) gives 
xn y? 
(Ae Fay aa 
that is so far as the second case is concerned all the rays that cut the plane of 
ay pass through the focal conic in that plane. 
Next, if we put y=0 we get alternatively 
P= 
ya Oy = Or Cee re es ae 1 
ee ae SS ae 
4 exh eae =o 5 : ‘ : (13) 
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