464 Cambridge Philosophical Society. 



and let 0^+0 y'=sO, 0„4-0j6'=O, b' being db: da. Hence 



a=A(s,y.y'). b=B(x.r/,y') ; x=X(a,b,b>), y=Y(a,b,b') 



the biordinal factors, y" —xi^'VyV')' b"—a(a,b,b'), disappearing 

 from 6' and y'. Hence 6' depends on a;, y, y'. Similarly, 6" depends 

 X, y,y',y', &c., and similarly for y',y",&c. If in_/'(.r,y,y',y",&c.) = 

 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 ■r=X, y = Y. 



13. There are two reciprocal biordinal equations belonging to the 

 modular equation ^(x, y, a, 6) = ; y"=x when a and b are constant, 

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

 of singular solution ; for Ay'(f>i, = Xbi(py. Let this be a(x, y, a, 6) = 0, 

 when cleared of y' or 6'. The following table exhibits the relations 

 of the double system : — 



I 4'{x,y,a,b)^=-0 1 



9{x,y,a,b)=0 \ <b^+(buy'=0 rf)a + (^A6'=0 | <T{x,y,a,b)=0 



I L. ^_L _, •, J r~V— -^ ,, 



il'=nx[x,y) a=A{x,y,y') b='R{x,y,y') x=X{a,b,b') y=\{a,b,b'] b=\{a,b) 



I ' -r -" ' -r -• I 



y=n(a;,C) y'—xKy.^') b"z=oi{a,h,b') b=\{a,Z). 



Eliminate a and b between ^=0, «r = 0, 0,,.|^ =0, and we have 

 y'=m, y=n, the singular primordinal and primitive of y"=x > those 

 of b"=za are obtained by eliminating x and y from (j) = 0, 0=0, 

 0t|j/=O. There is a relation involved between C and Z, the con- 

 stants of integration. For each value of C, y:=ll is the xy-curve 

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

 of Z and all values of «. The same of Z, b^A,a.nd(j>(x,n, a,b) = 0. 

 The contacts are of the second order, and y=n, 6 = A, are polar 

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

 primitives of y'=o; 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 (p(x,y,z,a)^0, \l>(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', 2')=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 = $(a', a, b, c) 2=^(a,', a, b, c), 

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



