POINTS OF THE INTEGRAL CALCULUS. [109] 



secting straight lines always make different angles on the same side with a third. Analysis, 

 without any such saving of extremes, launched itself into competition with the guarded 

 generalities of geometry ; of which we feel the effects to this day. We do at length admit 

 the necessity of a special inquiry into the cases of every formula which arise out of evan- 

 escence or infinitude : but we do not yet know the full interpretation of the maxim that 

 what is true up to the limit is true at the limit ; nor do we know whether this maxim is, in 

 any sense, universally true. Among the cases which offer difficulty, the following may be 

 noted. When there are phenomena, if such a term may be applied to mathematical results, 

 which are generally simultaneous, but which present occasional peculiarities, it has been 

 usually acted upon, if not expressly affirmed, that the criteria are either conclusive as to all, 

 or inconclusive as to all. That is, that if P and Q be two tests, of which Q is deducible 

 from P, each denoting usually the co-existence of the phenomena A, B, C, the exceptional 

 cases are all supposed to be equally excepted from both tests. But it may happen, that 

 while P continues, in the extreme and exceptional case, to point out the presence or absence of 

 A, without deciding on B and G, Q may decide on B or C without reference to A. 



When a family of curves, (p(x, y,c) = 0, is completely drawn, we may separate from among 

 the infinite number of curves which are equally related to all, the three following. First, a 

 curve of separation, over which when a point passes, the number of individuals of the family 

 which pass through the point undergoes an alteration. Secondly, a curve of contact, every point 

 of which is touched by one or more individuals of the family. Thirdly, a curve of resilience, 

 at every point of which an individual of the family stops and recedes, without intersection. 

 These three usually go together: as in the case of a family of equal circles with their centres 

 upon one straight line, to all of which a certain pair of parallel lines is a curve of separation, 

 contact, and resilience. But a curve may have one of the above names without the other two, 

 or two without the third. Thus the circles which touch an involute have contact without 

 resilience, and the involute may or may not be a curve of separation. And a locus of cusps 

 may be a curve of resilience without contact, and may or may not be a curve of separation. It 

 is also possible that one individual of the family itself may be a curve of either kind to the rest. 



The singular solution of a differential equation has been usually defined as the solution 

 which is not any case of the general solution. In this paper I propose to apply the term to 

 any solution whatsoever, be it contained in the general primitive or not, which results from any 

 process that cannot introduce an arbitrary constant : reserving the phrase extraneous solution 

 to signify any solution which is not a case of the general primitive. It has long been known 

 that cases of the complete primitive, when detected by the tests for extraneous solutions, fulfil 

 all the conditions of the latter, in a great many cases, with the single exception of the one from 

 which the name is derived. It is also well known that none of the ordinary modes of detecting 

 extraneous solutions are sufficient to separate them from the other singular solutions. 



Let the equation of the family of curves be (p(x,y, c) = 0, or c = <£(#, y) ; and let <!>, be 

 the partial* differential coefficient of <J> with respect to as, &c. 



I never use the symbol y, except where y is explicitly a \ dy . , , , „ , , . , 



f„n^ii/>^ „f „ tc . c _ i t •/, « -r must be employed, and it is -{y-x),-^(y-x),. 



function ot x. It y = x, for instance y,= 1: but if y-* = 0, dx ' * \a !• \n it 



