560 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
so that we may put 
CoC nce — Oe 
b', = (u— pr) C,, 0’, = (wu — pr) C,. 
Further, they give 
OM oM oM oM 
OPA © Cr+ serpy WO + 25 = DO reese a5 = 
whence, by comparison with (11), 
uK = (p— 2) pi. 
Again, since N = 0, 

el +5. eC +5, (= pe), + ter SS NO. =e. 
Multiply this by » — x and (10) by w and subtract. 
The equation connecting uv and v is then found to be 
fu—v(u—a)} (& Oy + 5 Cs = eee) 
Thus in general u—pv= —va, that is, cya@+b',=0=c,4+ b’, and the 
ditterential equations are satisfied. 
§ 48. In the case when N = 0, P + 0, we still have 
les Ok ee Or a Ch eeolby 2Gliny 
but the ratio de, : db, is unassigned. 
The equation to the tangent plane to the surface at 
(mM, ye + b,, Com + bg) 
oM oM 
(y, — aa — b,) a aL + (Y2 — 9 — bg) Ob 
DO) 
is, as always, 

Hence, when N = 0, the tangent plane at any adjacent point is 
me ‘oM, oM, \ lea (OM ie 
(y, — qa — b,) ( mm UC Oh } + (yz — cyv — by) (a, +d 2D, ) ai): 



Thus the tangent planes at all adjacent points pass through the same straight line 
y, = cx +), 
Yay = cx + by 
