WHICH SATISFY GIVEN CONDITIONS. 
119 
terms still more so ; but it is clear that the principle applies, and that the foregoing 
values, assuming them to be correct, would be obtained if only we know, for a unicursal 
curve without cusps, that 
0, b, c, . .)=( a +l)(b+l)(c+l) . . . [rm-(a+b+c , . .)]' 
( t the number of contacts a , b, c, . . .), and for a unicursal curve with a single cusp, 
that 
(a, b, c, . -.)= (a+l)(i+l)(c+l) • • • [rm-(a+b+c . . .) ]* 
— 2ja(5+l)(c+l) . . . [rm— (a+b+c . . .) — 1]* _1 , 
viz. that the diminution of a, b, c, . . .) occasioned by the single cusp is 
==[rm— {a+b+c, . . .)— l]* -1 .S{«(6+l)(c+l) . . .}. 
88. Consider a unicursal curve U“, and a curve O' having therewith t contacts of the 
orders a,b,c,... respectively. The coordinates (x, y, z) of any point of the unicursal 
curve are given as functions of the order roof a variable parameter 6 ; and substituting 
these values in the equation of the curve O, we have an equation of the degree rm in 6, 
but containing the coefficients of C r linearly ; this equation gives of course the values of 
Q which correspond to the rm intersections of the two curves. Hence in order that the 
curve C r may have the prescribed contacts with U m , the equation of the degree rm in Q 
must have t systems of equal roots, viz. a system of a equal roots, another system of b 
equal roots, &c. : this implies between the coefficients of the equation an(« + 5+c, . . .)- 
fold relation, which may be shown to be of the order («+l)(5 + l)(c+l). . . . 
[rm-{a-\-b-\-c, . . .)]*; and since the coefficients in question are linear in regard to the 
coefficients in the equation of the curve C r , the order of the relation between the last- 
mentioned coefficients has the same value ; that is, the number of the curves C r which 
have the prescribed contacts with the unicursal curve U m and besides pass through the 
requisite number of given points, is=(«-j-l)(5 + l)(c+l). . . . [rm— («+5+c, . . .)]'. 
89. The reduction in the case of a cusp appears to be caused as follows : — Consider on 
the curve U m a points indefinitely near to the cusp, and let the condition of the curve 
O' having the contact of the a - th order be replaced by the condition of passing through 
the a points ; that is, consider the curves O' which have with the curve U m (t— 1) contacts 
of the orders b, c, . . . respectively, which pass through the a points on the curve U m in 
the neighbourhood of the cusp, and which also pass through the requisite number of 
arbitrary points. The number of these curves is=(Z» + l)(c + l) . . . [rm — a — ( b-{-c -{- . .)]* -1 
(the term rm — a instead of rm, on account of the given a points on the curve : compare 
herewith He Jonquieees’ formula containing rm—p). Each of these curves, in that it 
passes through a points in the neighbourhood of the cusp, will ipso facto pass through 
a -\- 1 points (viz. a curve which simply passes through the cusp of a cuspidal curve meets 
the cuspidal curve there in two points, a curve which touches the cuspidal tangent meets 
the curve in three points, &c.), and be consequently, in an improper sense, a curve having a 
contact of the a-th order with the given curve U m . I assume that it counts as such curve 
a times, and this being so, we have, on account of the curves in question, a reduction 
