540 Mk DE MORGAN, ON SOME POINTS 



line of shading of this surface, the reciprocal a6c-surface has the original curve for one of 

 its lines of shading. 



Next, let the pole be placed on the surface c = /c(a, b). The polar arj/sf-surface is then sub- 

 ject to the primordinal equation C = k(A, B). If a and b be made compensating, by help of 

 <p a + d> c K a =0, <p b + <p c K b = o» substitution in <p(x, y, as, a, b, k) = 0, gives an #y#-surface reci- 

 procal to c = k (a, b), and subject to C = k{A, B). Let this surface be as = w (#, y) : then any 

 point on c = k has a polar surface which touches « = win a point ; and if this last point be 

 made a pole, its polar a&c-surface touches c = k in the original point. 



In the following sketch of the determination of the singular solution extreme cases, such as 

 arise from the absence of a variable, &c. are purposely omitted. A relation which makes any 

 differential coefficient of A, say A p , infinite, gives 



A x +Aj)+ A p r + A q s and A y + A z q + A p s + A q t 



proportional, independently of r, s, t, to 



A p, + A pzP + K r + A n s and A w + A pA + A vp s + A J- 



That is, A zt2>p>q and A p ^ p>q are proportional to A p{xiz>p>q) and A p{y{z>p>q) . If B p be also in- 

 finite, then A p and B p are related as if they were constant together. 

 We have then 



■ D P (i\z,p,q) _ Ii p(v\z,p,q) _ whence Jls,p ' 9 - ' |g '"'« = o 

 A A ' A A 



- a p<x\Z,p,q) P(ylz,p,q) • a -x\Z,p,q •^V\Z,p,q 



Hence the biordinal is satisfied by a singular solution obtained by making differential coeffici- 

 ents of A and B simultaneously infinite. As before, one general formula will suffice for in- 

 vestigation of properties. Since <p = 0, (j) z{z = 0, (p fit = are made identical by a = A, b = B, 

 c = C, the equations on the left determine A p , B p , C p . (Remember that <p p = 0, <p pltlel = <p z , 



+ <Pa A P + <P b B p + <Pc C P = 0.(^ te |0 W |-0„i^|) 



<Pz + <j>aMz) A p + <P»(*\Z) B P + 0e(xl,)^p = ° + &(&»I0.»I " 0„|0 W |) = 0. 



+ <?Wu>4> + ^wb- 8 , + 0c<„U) C p = ° + ^(iiii - jrViM 



On the right is the ordinary condition under which A p , B p , C p , become infinite : this is 

 the condition of singular solution answering to <p a (p bxl — <p b <p ax \ = in equations of two varia- 

 bles. By eliminating a, b, c, between this condition and = 0, <p x[z = 0, <p yU = 0, we obtain 

 a primordinal singular solution. The ordinary primitives of this satisfy the biordinal : not so 

 the singular primitive, obtained from <p = 0, <p a — 0, <p b = 0, (p c = 0. 



If a, b, c, be now treated as variable, and a, y, %, as constants, correlative primordinal 

 and biordinal equations may be obtained. The criterion of singular solution is the same for 

 both equations. In the criterion obtained above, write the expansion of each complex term, 

 as <p bx + <p bz p for <p bxt : substitute - <j>/-<p z and - (p y -(p z for p and q, and we shall obtain 

 T= 0, T consisting of nine terms, of which a 0, (0 6 „0 re - 4MM and $.$,(&&, - (p hx <pj 



