Prof. Cayley on Differential Equations and Umbilici. 375 



general a nodal curve on the surface, and it may be spoken of as 

 the nodal curve ; the latter of them is the reduced or proper 

 envelope, or simply the envelope. And the terms nodal curve 

 and envelope may also be applied to the curves Q=0 and □ =0, 

 which are the projections on the plane of xy of the first-men- 

 tioned two curves respectively. There is however a case of 

 higher singularity which it is proper to consider. Suppose that 

 Q and □ have a common factor K, say Q = KR, □ =KV- The 

 complete envelope Q 2 D =R 2 K 3 V=0 here breaks up into the 

 nodal curve R = twice, the cuspidal curve K = three times, 

 and the reduced or proper envelope V = once. 



Reverting for a moment to the form (z + X)(z-{-Y) = 0, 

 the derived equation Xl = -(X-Y) 2 X'Y'=0 is satisfied by 

 (X— Y) 2 =0, this equation, or say the equation of the envelope, 

 being in fact the singular solution of the differential equation. 

 This assumes however that the differential equation is given in 

 the form in which it is immediately obtained by derivation from 

 the integral equation, without the rejection of factors which are 

 functions of the coordinates (x, y) only. It is proper to con- 

 sider the reduced equation obtained by rejecting such factors. 

 Thus if X and Y are rational functions, the reduced form is 

 X'Y^O, which is no longer satisfied by the equation (X — Y) 2 = 0. 

 In the before-mentioned case where the roots are P + Q s/ □ 

 (or (X— Y) 2 =Q 2 Q), P, Q, and □ being rational functions of 

 [pCy y), the derived equation 



o = -Q 2 {4nP' 2 -(2Q'n+Qn') 2 }=o 



divides out by the factor Q 2 , but it does not divide out by □ ; 

 the reduced form is therefore 



4nP /2 -(2Q'n+Qn') 2 =(), 



which is not satisfied by Q=0, while it is still satisfied by □ =0 

 (since this gives also D f =0) ; that is, the nodal curve Q = is 

 not a solution of the differential equation, but we still have the 

 singular solution □ =0, which corresponds to the reduced or 

 proper envelope. In the case Q=KR,D=KV of a cuspidal 

 curve, the above form of the differential equation becomes 



4KVP' 2 - -|3KK'RV+ K 2 (2V"&' + V'R)^ 2 =0, 



which divides out by K ; and when reduced by the rejection of 

 this factor, it is no longer satisfied by the equation K = 0, which 

 belongs to the cuspidal curve ; that is, neither the nodal curve 

 R = nor the cuspidal curve K = is a solution of the differen- 

 tial equation, but we still have the singular solution V = 0, which 

 corresponds to the reduced or proper envelope. It would appear 



