OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 541 
Thus the lines of any congruency satisfy differential equations of CrLATRAuT’s 
form. 
§ 28. If two consecutive lines meet, the equations (1) and (2), with 
O) aioe Cle le ekg eh 5. 2) 5 8) (8), 
DS ava fas eae ein Ve Oe, is oo pe a (A) 
must form a consistent system. 
Hence 
HO Okey —= Wheaten sO) oe ded 5 ge gS) 
This equation gives two values for the ratio dc, :dc,, and shows, therefore, that 
each line of the system meets two consecutive lines (SALMon, “ Geometry of Three 
Dimensions,” § 456). 
We are supposing that b,, b, are regarded as functions of ¢,, cg. 
§ 29. The elimination of ¢, ¢, de,, de, from the equations (1), (2), (3), (4) will give 
the equation to a surface, and the tangent plane to this surface will contain the line 
(1), (2). For the equation to the tangent plane is found by eliminating dc, dc,, dr 
from the differentials of (1) and (2) and of 
gare Ue 
DN lrromant aa Ons ey DORM ee Tes sree WO) 
ab, ab\ 
and substituting X — a, Y, — y,, Y,— y, for dx, dy,, dy, (6) and (7) are found 
from (3) and (4) by the use of the undetermined multiplier X. 
From (1) and (2) we have, considering c, and c, and therefore also b, and bd, as 
functions of x, ¥;, Yo; 
dy, = c,dx + «dc, + db,, 
dy, = Cy da + «de, + dbsg, 
and therefore in virtue of (6) and (7) 
dy, + \ dy. = (ce, + Ac.) da. 
The equation to the tangent plane is therefore 
Y¥, — |X +A(Y, — @X) = y, — oy" +A (Ya — C0) 
= b, + Aby, 
and the tangent plane contains the line (1), (2). 
