140 



from b' and y'. Hence b' depends on x, y, y'. Similarly, b" depends on 

 x, y, y', y", &c, and similarly for y',y",&.c. If in f(x,y, y',y",&.c.)=Q 

 we substitute for x, y, y', &c. in terms of a, b, b', &c, the two equa- 

 tions belong to polar reciprocals. If either can be integrated, the 

 integration of the other depends on elimination : thus if the equa- 

 tion in a, b, &c. can be integrated, the solution of the equation in 

 x, y is obtained by eliminating a and b between the integral obtained 

 and x=X, y=Y. 



13. There are two reciprocal biordinal equations belonging to the 

 modular equation <p(x, y, a, b) = ; y"=x wnen a an( ^ & are constant, 

 b"-=s.a. when x and y are constant. The two have the same condition 

 of singular solution ; for Ay^j=Xj/^. Let this be a{x, y, a, b)=0, 

 when cleared of y' or b'. The following table exhibits the relations 

 of the double system : — 



- 4>(x,y,a,b)=0 — - — , 



<r(x,y,a,b)=0 d> x +cf)yy'=0 $a+$fi&— \ (r{x,y,a,b)—0 



| L , 1 , , 1 -,—J 1 



y'=-us[x,y) a=A(x,y,y') b=B(x,y,y') x= X(a,b,b') y =Y(a,b,b') b'=\(a,b) 



y=U(x,C) y"=x(*,y,y') b"=*{a,b,V) b=A(a,Z). 



Eliminate a and b between 0=0, 17=0, fa y =0, and we have 

 y' = zff, y=n, the singular primordinal and primitive of y"=-%; those 

 of 6"=a are obtained by eliminating x and y from 0=0, <r=0, 

 (h x y=0. There is a relation involved between C and Z, the con- 

 stants of integration. For each value of C, y = n is the a?y-curve 

 which touches all in (p(x,y, a, A) = 0, for the corresponding value 

 of Z and all values of a. The same of Z, b= A, and <p(x, n, a, b)=0. 

 The contacts are of the second order, and y=U, b=A, are polar 

 reciprocals for corresponding values of C and Z. But the singular 

 primitives of y'=in and b'=X are not necessarily reciprocals: when 

 this does happen, their contacts with primitives are of the third 

 order. 



14. When a surface is described by one set of curves, as in the 

 surface obtained by eliminating a from <f>(x,y,z,a)=0, \jj(x,y,z,a)=0, 

 it is proposed to call it a shaded surface, and the curves lines of sha- 

 ding. The equation f(x, y, z, y', z')=0, y and z being functions of 

 x, cannot, generally, belong to any family of surfaces in an unre- 

 stricted sense ; that is, it cannot be always true of a point moving 

 in any way upon a surface. Such a supposition would be equivalent 

 to imagining a surface every point of which has the primordinal 

 character of the vertex of a cone. But it may belong to any surface, 

 properly shaded, or to any mode of shading, if the proper surface be 

 chosen. 



15. Two equations of the form y = $(x, a, b, c) z=^(x, a, b, c), 

 give one, and only one, primordinal of the form/(#, y, z,y',z') = 0. 

 Assume any surface w(x,y, z)—0; by this, and compensative rela- 

 tions between a, b, c, another pair of primitives may be found. But 

 the primitives obtained from w=0 do not shade this surface, except 

 in cases determined by two relations between the constants. Again, 

 making a, b, c compensative, without any assumed surface, we find 



