MINDING’S SYSTEM OF FORCES. 529 
All the rays must cut the plane of yz, let us therefore put a=0, and con- 
sider where the rays through (0dc) cut the plane of zy. Putting z=0, and Minding's 
Theorem 
substituting for \,y7,, 7° being now b’+c’, we find that, if we restrict ourselves whe /=°. 
to Minp1N@’s case by putting f=0, (25) and (26) become 
—(P7VP+V0)\7—-(7P+V)ey+gi(y—by+hie=0, . (27) 
and —rxv?—Cy +P(y—bYt+VC=0. : ‘ ' (28) 
(27) and (28) intersects in the points in which the rays through (Odc) cut yz. 
But, if we multiply (28) by g’, and subtract (27) from it, we get 
Ae -M\eLhg ee =0. . 9. . (29) 
Hence, if c be not =0, all the points in which rays cut zy lie on the conic 
a Bret. 
ri — |) Pola . . . (30) 
Similarly, if b be not =0, all the points in which rays cut xz lie on the conic 
reese oe , (31) 
It will be found that the cases c=0, 6=0 lead only to particular cases of 
the general theorem. 
Hence the congruency is identical with the doubly infinite system of rays the congru- 
ency is of the 
that intersect the focal conics of the ellipsoid fourth class. 
ge 
y” 7g ca 1 4 
+h LE hz at Gray. cy e ° ° (32) 
it is, therefore, of the fourth class. 
Returning to the more general case, if we denote /’€? + 9’n’ + WC? by gq’, (8) 
and (9) may be written 
(PP I+ grt (—Wy=0 2 6. (88) 
(p§ + g'— Sry + (p? +g — Gur’ + (pi tg —AApy=0. 2. (34) 
Whence 
AV=P{ ptt gy —(P+M)p +97} (9? —h) 
by =P 4 &e. } 
Cid oa ghd ll ie : (35) 
