141 



one equation of the form (a, b, c, a', b')=0, any primitives of which 

 lead to other primitive forms for/=0. Each of the second primi- 

 tives has contact of the first order with one family of curves from 

 among the original primitives ; and all ordinary primitives are found, 

 in an infinite number of ways, among the connecting curves of others. 

 There is a singular solution, a curve of contact to all primitives, when 

 <J> a =0, ^ a =0, &c. can all be satisfied at once. 



Since y=<&, s=^, give a primordinal equation independent of 

 constants, the polar reciprocal properties of curves in space are of a 

 restricted form. Every surface dictates another surface, and a mode 

 of shading both, so that each line of shading on either surface is the 

 polar reciprocal of a line on the other. 



16. The conversion of constants into compensative variables may 

 give restricted solutions, as in the ordinary case of two variables, and 

 every other in which the constants are converted into separately 

 self- compensating variables. When these variables are made collect- 

 ively compensating, and the equations permit elimination of the 

 original variables, ordinary differential equations may be produced, 

 the integration of which may, after substitution, give primitives of 

 the same form as those from which they came. But when the ori- 

 ginal variables cannot be eliminated, arbitrary relations may be 

 required, in number enough to eliminate the differentials of the new 

 variables : in this case arbitrary functions enter the primitives finally 

 deduced. Of this last case one instance is Lagrange's transition 

 from a primitive of a primordinal partial equation having two con- 

 stants to the complete primitive of that equation. 



17. A biordinal partial equation may be produced from 



\](x, y, z, a, b, c, e, h)=0 



by eliminating the five constants between U=0 and the five results 

 of primordinal and biordinal differentiation. But it is not true that 

 every form of U=0 leads to one biordinal equation only : many 

 forms lead to an infinite number. Two attempts to procure other 

 primitives by making a, b, &c. compensative variables, end in two 

 different forms of result. First, when all the resulting equations 

 are required to be integrable, by introduction of a proper factor, the 

 success of the process requires the integral of two partial equations, 

 one primordinal and one biordinal, between four variables. Secondly, 

 when no such condition is required, the result is another form in- 

 volving five constants. 



18. A primordinal partial equation belongs to a family of surfaces 

 of which one is determined by any given curve through which it is 

 to be drawn. A biordinal equation belongs to an infinite number of 

 families ; and a distinct conception of the conditions which select 

 an individual surface is best formed by an extension of the following 

 kind. A curve on a surface is analogous to a point on a curve : two 

 curves being drawn on a surface, the analogue of the chord joining 

 two points on a curve is the developable surface (or surfaces) drawn 

 through the two curves. The developable surface which touches 

 the given surface in a curve (and not the tangent plane) is the ana- 



